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     {\huge СИБИРСКИЕ \ ЭЛЕКТРОННЫЕ} \\[2mm]
     {\huge МАТЕМАТИЧЕСКИЕ ИЗВЕСТИЯ} \\[2mm]
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        {\rm\small DOI~10.33048/semi.2019.16.xxx}\hphantom{aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa}
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\begin{document}
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\title[Gradient flow for Kohn-Vogelius functional]{Gradient flow for Kohn-Vogelius functional}
\author{{P.I. Plotnikov}, {J. Sokolowski}}%
\address{Plotnikov P.I.
\newline\hphantom{iii} Lavrentyev Institute of hydrodynamics RAS
\newline\hphantom{iii} pr. Lavrentyeva, 15,
\newline\hphantom{iii} 630090, Novosibirsk, Russia}
\email{piplotnikov@mail.ru}

\address{Sokolowski J.
\newline\hphantom{iii} Systems Research Institute of the
Polish Academy of Sciences,
\newline\hphantom{iii} ul. Newelska 6, 01- 447 Warszawa,
Poland,
\newline\hphantom{iii}Institut Elie Cartan, UMR 7502 Laboratoire de Mathematiques,
\newline\hphantom{iii}Universite de Lorraine,
\newline\hphantom{iii} Nancy 1, B.P. 239, 54506 Vandoeuvre
Les Nancy Cedex, France,
\newline\hphantom{iii}Department of Scientific Computing, Informatics Center,
\newline\hphantom{iii}Federal University of Paraiba,
\newline\hphantom{iii} 471 Rua dos
Escoteiros s/n, Mangabeira,
\newline\hphantom{iii}Joao Pessoa,
Paraiba 58058- 600, Brazil. } \email{
Jan.Sokolowski@univ-lorraine.fr}









%\thanks{\sc Plotnikov P.I.}
%\thanks{\copyright \ 2021 Плотников П.И.}
%\thanks{\rm Работа поддержана РНФ(грант 15-11-20019)}\thanks{\it Поступила  г., г.}%

%\author{Sokolowski J.}



 \semrtop \vspace{1cm} \maketitle {\small
\begin{quote}
\noindent{\sc Abstract. } The identification problem of an inclusion
is considered in the paper. The inclusion is unknown subdomain of a
given physical region. The available information on the inclusion is
governed by measurements on the boundary of this region. In
particular, the single measurement problem of impedance
electrotomography and similar inverse problems are included in our
approach. The shape identification problem can be solved by the
minimization of an objective function taking into account the
measurement data. The best choice of such objective function is the
Kohn-Vogelius energy functional. The standard regularization of the
Kohn-Vogelius functional include the perimeter and Willmore
curvature functional evaluated  for an admissible inclusion
boundary. In the two-dimensional case, a nonlocal existence theorem
of strong solutions is proved for the gradient flow dynamical system
generated for such a regularization of the Kohn-Vogelius functional.
Bibliography: 24 titles.
 \noindent

{\bf Keywords.} Shape optimization, inverse problems, Willmore flow,
Euler elastica.
 \medskip

\noindent{\bf Keywords:} Shape optimization, inverse problems,
Willmore flow, Euler elastica.
 \end{quote}
 \tableofcontents


\section{Introduction}\label{preface}
\renewcommand{\theequation}{1.\arabic{equation}}
\setcounter{equation}{0}This paper is devoted to applications of the
theory of geometric flows to shape optimization problems. The
beginning of the modern mathematical theory of shape optimization
was laid in monographs \cite{SOKZOL}, and \cite{DELZOL}. In the
monographs, it was first singled out as an independent scientific
discipline. At present, the theory of shape optimization includes a
large number of various applied problems.
%We focused on the model

In this paper we deal with basic $2D$  shape optimization problem
which admits the following  formulation. Fix an arbitrary bounded
simply connected domain  $\Omega\subset\mathbb R^2$. It is supposed
to contain the inclusion $\Omega_i$ such that
$\overline{\Omega_i}\subset \Omega $. The shape of the inclusion is
unknown and must be determined together with the solution of the
boundary value problem. Let a Jordan curve $\Gamma$ be the boundary
of $\Omega_i$. In this setting the interface $\Gamma$ split $\Omega$
into the inclusion $\Omega_i$ and the curvilinear annulus
$\Omega_e=\Omega\setminus\overline{\Omega_i}$. Finally, fix an
arbitrary constant $a_0>0$ and set
\begin{equation}\label{preprint1}
a(x)=a_0\text{~~in~~}\Omega_i, \quad a=1\text{~~in~~}\Omega_e.
\end{equation}
As a basic example, we consider  the single measurement
identification problem arising in the electrical impedance
tomography, \cite{ADK}. Electrical impedance tomography is used in
medical imaging to reconstruct the electric conductivity of a part
of the body from measurements of currents and voltages at the
surface. The problem can be formulated as follows:

For given $g,h:\partial\Omega\to \mathbb R$ satisfying the condition
\begin{equation}\label{gala4}
g\in L^2(\partial \Omega), \quad h\in W^{1/2,2}(\partial\Omega),
\quad \int_{\partial\Omega} g\, ds=0
\end{equation}
it is necessary to find $\Gamma$ and an electric potential
$u:\Omega\to \mathbb R$ satisfying the equations
\begin{equation*}
\text{div~}(a\nabla u)=0\text{~~in~~}\Omega, \quad a\nabla
u\cdot\boldsymbol\nu=g, \quad u=h\text{~~on~~}\partial\Omega,
\end{equation*}
where $\boldsymbol\nu$ is the outward normal vector to
$\partial\Omega$. More generally, the problem of identification is
to determine the shape of an inclusion by the additional boundary
condition. This inverse problem is ill-posed and in general case has
no solution. Its approximate solution can be found by using the
shape optimization approach. Thus we come to the following
variational problem. Denote by  $v,w:\Omega\to\mathbb R$ the
solutions to boundary value problems
\begin{align}
\text{div~}(a\nabla v)  & = 0 & \text{div~}(a\nabla w)  & = 0 &
\text{in~~} & \Omega,
\\   \label{shopt111}
a\nabla v\cdot  \boldsymbol\nu & = g & w & = h & \text{on~~} & \partial\Omega, \\
 & \int_{\partial\Omega} v\, dx=0. &  &
\end{align}
Next,   define a positive  objective function that vanishes if and
only if $v=w=:u$. The most successful choice of the objective
functional is the Kohn-Vogelius energy functional, which is given by
the formula, \cite{kohnvog},
\begin{equation}\label{shopt2}
    J(\Gamma)=\int_\Omega a\nabla(v-w)\cdot\nabla(v-w)\,dx.
\end{equation}
Note that for fixed $h$ and $g$, it depends only on $\Gamma$.




 Unfortunately,   shape optimization problems as stated  with no additional geometric constrains
 are  ill-posed, see
 \cite{murat}, \cite{TARTAR} for examples.
The reason is that  microstructures tend to form, which are
associated with a weak convergence of the characteristic functions
along a minimizing sequence $\Omega_i^m$, $m\geq 1$. Indeed, in the
absence of strong compactness of the minimizing sequences of
designs, the optimal state should be attained by a fine mixture of
different  phases.

The widely used  method to cope with these difficulties is to
penalize the shape perimeter by adding a regularizing term  to the
objective functional:
\begin{equation}\label{perimeter1}
    \epsilon_p \,\mathcal L+J.
\end{equation}
Here $\mathcal L$ is the perimeter of $\Omega_i$, $\epsilon_p>0$ is
the regularization
 parameter. If $\Gamma=\partial \Omega_i$ is a regular curve,
then $\mathcal L$ is  the length of $\Gamma$. This penalization was
proposed in \cite{AMB} by analogy with the Mumford-Shah functional,
\cite{MUMFORD}, in the theory of image segmentation processes.  The
stronger regularization may be obtained if we impose constrains on
the curvatures of $\Gamma$. This approach also was motivated by the
theory of image processing, \cite{MUMFORDEL}. The only possible
geometrically invariant penalization functional depending on
curvatures is  the 1-dimensional Willmore functional  (Euler
elastica) defined by the equality
\begin{equation}\label{WIILMORE1}
\mathcal E_e(\Gamma)\,=\, \frac{1}{2}\int_\Gamma |k|^2\, ds,
\end{equation}
where $k$ is the  curvature vector of $\Gamma$. Therefore, we can
define the strong regularization of an objective function as follows
\begin{equation}\label{penalstrong1}
     \mathcal E+J, \text{~~where~~}\mathcal E=\epsilon_e\,\mathcal E_e+\epsilon_p\,\mathcal
     L.
\end{equation}
Here $\epsilon_j$, $j=e,p$, are some positive constants. Note that
the penalization term  can be interpreted as the cost of structure
manufacturing. Hence $\epsilon_j$, $j=e,p$, are not
%necessary
supposed to be small. Without loss of generality we will assume that
$\epsilon_j=1$, which leads to the following expression for
$\mathcal E$
\begin{equation}\label{penalstrong}
\mathcal E=\int_\Gamma\big(\, \frac{1}{2}|k|^2+1\big)\, ds=\mathcal
E_e+\mathcal
     L.
\end{equation}
The most important question  of the theory is the construction of a
robust algorithm for the numerical study of
 shape optimization problems.  The standard approach is to use the steepest
 descent method based
on the shape calculus developed by Sokolowski and Zolesio (1992),
\cite{SOKZOL}. See also Delfour and Zolesio (2001), \cite{DELZOL},
and references therein. The shape calculus works for inclusions $
\Omega_i$ with the regular boundary $\Gamma=\partial \Omega_i $. In
this setting, the objective  function $J$ is considered as a
functional defined on the totality of smooth  curves $\Gamma$. This
assumption is natural from the practical point of view. Without loss
of generality we may restrict our considerations by the class of
twice differentiable immersions (parametrized curves) $f:\mathbb
S^{1}\to \mathbb R^2$ with $\Gamma= f(\mathbb S^{1})$ diffreomorphic
to the circle $\mathbb S^{1}$. In this framework, we will use the
denotation $J(f)$ along with the denotation $J(\Gamma)$. The main
goal of the shape calculus is to develop the method of
differentiation of objective functions with respect to shapes of
geometrical objects.


The shape derivative of an objective function is defined as follows.
Choose an arbitrary vector field $X:\mathbb S^{1}\to\mathbb R^2$ and
consider the immersion
$$
f^t(\theta)=f(\theta)+tX(\theta), \quad t\in (-1,1), \quad \theta\in
\mathbb S^{1}.
$$
The curves $\Gamma^t=f^t(\mathbb S^{1})$, $t\in (-1,1)$, determine
1-parametric family of perturbations of $\Gamma$. The shape
derivative $\dot{J}$ of  $J$ in the direction $X$ is defined by the
equality
\begin{equation}\label{gala9}
\dot{J}(\Gamma)\,[X] \,=\,
    \frac{d}{dt}\, J(\Gamma^t)\Big|_{t=0}.
\end{equation}
If it admits the Hadamard representation
\begin{equation}\label{eulervar1}
\dot{J}(\Gamma)\,[X] \,=\,\int_\Gamma \phi\,  n\cdot X\, ds,\,
\phi\in L^1(\Gamma),
\end{equation}
where $n$ is the inward normal to $\Gamma=\partial\Omega_i$, then
the vector field
\begin{equation}\label{eulervar2}
dJ(\theta):=\phi(\theta)n(\theta), \quad \theta\in \mathbb S^{1},
\end{equation}
is said to be the gradient of $J$ at the point $f$.

For the transmission single measurement identification problem,
 the gradient $dJ$ of the Kohn-Vogelius objective function
\eqref{shopt2} is defined by the equality, see \cite{ADK},
\begin{equation}\label{gala11}
d J= 2\big(a\partial_n v\big[\partial_n v\big]- a\partial_n
w\big[\partial_n w\big]) n-\big[a\nabla v\cdot\nabla v-a\nabla
w\cdot\nabla w\big] n,
\end{equation}
where $v$, $w$ are solutions to problem \eqref{shopt111},
$\big[\cdot\big ]$ denotes the jump across $\Gamma$.










The similar definition holds for the geometric energy functional
$\mathcal E$ (see Lemma \ref{eulervar3} in Section \ref{results}).
Note that that the shape gradient can be regarded as a normal vector
field on $\Gamma$.


 If $f$ is sufficiently smooth, for example
$f\in C^{4}$, then the mapping $f-\delta\, d(\mathcal E+J)\,(f)$
defines an immersion of $\mathbb S^{1}$ into $\mathbb R^2$ for all
sufficiently small $\delta>0$. In the steepest descent method, the
optimal immersion $f$ and the corresponding shape $\Gamma=f(\mathbb
S^{1})$ are determined as a limit of the sequence of immersions
\begin{equation}\label{gala13}
    f_{n+1}=f_n-\delta\,\, \big(\, d\mathcal E(f_n)+ d J(f_n)\,\big),
    \quad n\geq 0,
\end{equation}
and the corresponding sequence of curves $ \Gamma_n=f_n(\mathbb
S^{1})$. Here the energy $\mathcal E$ is defined by
\eqref{penalstrong}, $\delta$ is a fixed positive number, usually
small, $f_0$ is an arbitrary admissible initial shape. Relation
\eqref{gala13} can be considered as the time discretization  of the
Cauchy problem
\begin{equation}\label{gala14}
    \partial_t f(t)= -\big(\, d\mathcal E(f(t))+d J(f(t))\,\big),\quad
    f(0)=f_0.
\end{equation}
Note that here $t$ is an artificial quasi-time related to the
steepest descent method.

Since $\mathcal{E}(f(t))+J(f(t))$ is a decreasing function of $t$, a
solution to problem \eqref{gala14} can be considered as an
approximate solution to the penalized variational problem
$$
\min\,\big( \mathcal E+J\big).
$$
Hence the existence of a solution to Cauchy problem \eqref{gala14}
guarantees the well-posedness of the steepest descent method. In its
turn, the existence of the limit $\lim\limits_{t\to \infty} f(t)$
guarantees the convergence of the method. Hence the  task is to
investigate the well posedness of Cauchy problem \eqref{gala14}. The
main goal of this paper is the proof of the existence of global
smooth  solution to the Cauchy problem \eqref{gala14} for an
arbitrary smooth initial data, see Section \ref{results}.
 The paper is organized
as follows.\\
\noindent In Section \ref{preliminaries},  we give  basic
definitions and characterize the elementary properties  of curves
with finite elastic energy. We also collect the basic facts from the
theory of
Sobolev spaces, which will be used throughout the paper.\\
 \noindent In Section \ref{results}, we formulate the main result
on existence of  global solution to Cauchy problem \eqref{gala14}.
We give the outline of the proof and formulate the main a priori
estimates of solutions. Furthermore, we consider in details the
dependence of smoothness of the gradient $dJ$ on the curvature of
the interface $\Gamma$. In Section \ref{gradiend}, we give the proof
of main estimates for the gradient of Kohn-Vogelius functional. In
Sections \ref{appendixb} and \ref{section7}, we give the proof of a
priori estimates for the gradient flow of penalized Kohn-Vogelius
functional. In Sections \ref{lemmageom}--\ref{APPE}, we collect
auxiliary results which are used throughout of the paper.










\section{Preliminaries}\label{preliminaries}
\renewcommand{\theequation}{2.\arabic{equation}}
\setcounter{equation}{0} In this section we collect the basic facts
on the theory of planar curves and the theory of Sobolev spaces on
the real axis.


 \subsection{Geometric lemmata}\label{notation0}
  Further we will
consider special class of immersions $f:\mathbb S^1\to \mathbb R^2$
satisfying the conditions
\begin{equation}\label{immer1}
    \int_\Gamma \big(\frac{1}{2}|k|^2+1\,\big)\, ds\leq E_0, \quad \Gamma=f(\mathbb
    S^1).
\end{equation}
Our considerations are based on the following elementary  lemmas on
the properties of such immersions. The first gives the double-side
estimates for the length $\mathcal L$ in terms of the energy bound
$E_0$.

\begin{lemma}\label{bokr2}
The estimate
\begin{equation}\label{bokr3}
    \frac{2}{E_0}\leq \mathcal L\leq E_0
\end{equation}
holds true for every curve $\Gamma$ satisfying condition
\eqref{immer1}.
\end{lemma}
\begin{proof} The proof is given in Section \ref{lemmageom}.
\end{proof}
 The second lemma provides the local graph representation of planar curves with
square integrable curvature. Let us consider the following
construction.

 Choose an arbitrary immersion satisfying condition \eqref{immer1}. Let $z=f(\theta_z)\in \Gamma$ be an arbitrary point. Fix arc-length coordinate
$s$ such that
$$
s(z)=0\text{~~and~~}-\mathcal L/2\leq s<\mathcal L/2.
$$  For every $0<\kappa<\mathcal
 L/2$, denote by $\Gamma_\kappa$ the arc
 $$
 x= f(s), \quad -\kappa<s<\kappa.
 $$
 Next, introduce the Cartesian coordinates $(x_1,x_2)$ with origin at
$z$  such that the axis of abscissa
 is directed along the tangent vector $\tau(\theta_z)$ and the axis
 of ordinate is directed along the normal vector $
 n(\theta_z)$. The consequent  results are independent  of  choice of
 $z$.
Now, our task is to show that the curve $\Gamma$ is locally
%can be
represented as a graph of $C^{1+\alpha}$ function in a neighborhood
of the origin $z$.








\begin{lemma}\label{bokr3a}Under the above assumptions, there exist
positive numbers $\kappa$, $\alpha$, $\beta$, and $c$, depending
only on the constant $E_0$ in \eqref{immer1}, and  the function
$\eta\in C^1(-\alpha, \beta)$, $\eta(0)=0$, with the following
properties
\begin{equation}\label{bokr4}\begin{split}
    0<c^{-1}\leq \kappa, \alpha, \beta\leq c<\infty,\\
    \| \eta'\|_{C(-\alpha, \beta)}\leq 1/6, \quad\|\eta''\|_{L^2(-\alpha, \beta)}\leq
    c\|k\|_{L^2(\Gamma_{3\varkappa})}.
\end{split}\end{equation}
Here $\eta'(x_1)=\partial_{x_1}\eta(x_1)$. Moreover, the mapping
$x_1\to (x_1, \eta(x_1))$ defines $C^1$-parametrization of the arc
 $\Gamma_{3\kappa}$ and takes diffeomorphically the interval $(-\alpha, \beta)$
 onto this arc.
\end{lemma}

\begin{proof} The proof is given in Section \ref{lemmageom}.
\end{proof}






 Lemma \ref{bokr3a} gives the simple criterium on the absence of
 self intersections of curves $\Gamma$ satisfying the energy
 condition \eqref{immer1}.

\begin{corollary}\label{bokr3s} Let an immersion $f:\mathbb S^1\to\mathbb R^2$ meets all
requirements of Lemma \ref{bokr3a}.  Furthermore assume that there
is $\nu>0$ with the property
\begin{equation}\label{immer23s}
    \text{\rm dist~}(\Gamma\setminus \Gamma_{3\kappa}, \,
    \Gamma_{2\kappa})\,\geq \nu.
\end{equation}
Then $\Gamma$ has no self-intersections. Conversely, if~ $\Gamma$
has no self-intersections, then inequality \eqref{immer23s} holds
for some $\nu>0$.
\end{corollary}
\begin{proof} The corollary is an obvious consequence of Lemma \ref{bokr3a}.
\end{proof}

The second corollary extends the previous results to the case of
families of immersions with finite elastic energy. Let us consider a
family of immersions $f(t,\cdot):\mathbb S^1\to\mathbb R^2$, $t\in
[0,T]$. Every immersion $f(t,\cdot)$, satisfying condition
\eqref{immer1}, defines $\mathcal L(t)$- periodic function of the
arc-length variable $s$,
$$
\overline{f}(t,s)= f(t,\theta(s)).
$$
Note that the periods  $\mathcal L(t)$ are uniformly bounded from
below and above by the constants $2/E_0$ and $E_0$. Moreover, the
functions $\partial_s^2 \overline{f}(s,t)$ are uniformly bounded in
$L^2(-\mathcal L(t)/2, \mathcal L(t)/2)$. It follows that the set of
the mappings $\overline{f}(t,\cdot)$, $t\in [0,T]$, satisfying
\eqref{immer1}, is relatively compact in $C^1(\mathbb R)$.

Assume that a family of immersions $f(t)$, $t\in [0,T]$, satisfies
the following conditions
\begin{itemize}
\item[\textbf{ G.1}] The   curves $\Gamma(t)=f(t,\mathbb S^1)$ have no
self-intersections.

\item[\textbf{ G.2}] The immersions $f(t)$ satisfy energy condition
\eqref{immer1} with the constant $E_0$ independent of $t$.

\item[\textbf{ G.3}] The set of the mappings $f(t,\cdot)$, $t\in [0,T]$ is
compact in the space $C(\mathbb S^1, \mathbb R^2)$.

\end{itemize}

It follows from Lemma \ref{bokr3a} that for every $f(t, \theta)$,
$t\in (0,T)$, there is $\kappa\in (0, 2/E_0)$ which meets all
requirements of this lemma and is independent of $t$.



\begin{corollary}\label{bokr3st} Let a family of immersions  $f(t, \cdot):\mathbb S^1\to\mathbb R^2$
satisfies conditions {\rm \textbf{ G.1}}-{\rm \textbf{ G.3}}.  Then
there is $\nu>0$ such that
\begin{equation}\label{immer2g}
    \text{\rm dist~}(\Gamma(t)\setminus \Gamma_{3\kappa}(t), \,
    \Gamma_{2\kappa}(t))\,\geq \nu
\end{equation}
for all $t\in [0,T]$ and for all arcs $\Gamma_{3\kappa}(t)$ given by
Lemma \ref{bokr3a}.
\end{corollary}
\begin{proof} The proof is given in Section \ref{lemmageom}.
\end{proof}
\subsection{Function spaces}\label{sobolevfun}
\paragraph{Sobolev spaces of periodic functions.} For every integer $r\geq
0$, denote by $H^r_\sharp$, the Sobolev  space of all $\mathcal L$
-periodic mappings with the finite norm
\begin{equation}\label{appb7}
    \|f\|^2_{H^r_\sharp}=\int_0^{\mathcal L} (|f|^2+|\partial_s^r f|^2)\, ds.
\end{equation}
For real $r\geq 0$, the space $H^r_\sharp$ is defined by the
interpolation. Note that  the equivalent norm in $H^r_\sharp$ may be
defined by the equality
\begin{equation*}
 \|f\|^2_{H^r_\sharp}=\sum_{m\in\mathbb Z}(1+|m|^{2})^r |f_m|^2,
\end{equation*}
where the Fourier coefficients
$$
f_m=\frac{1}{\sqrt{\mathcal L}}\int_0^{\mathcal L}
e^{-\frac{2\pi}{\mathcal L}m \,s\, i}\,\, f(s)\, ds.
$$
If $\Gamma$ is a rectifiable Jordan curve of the length $\mathcal
L$, then the curvature of $\Gamma$, the gradient of Kohn-Vogelius
functional, tangent and normal vectors of $\Gamma$ can be regarded
as $\mathcal L$-periodic functions of the arc-length variable $s$.
By this reason, we will use the parallel denotations for
$H^r_\sharp$:
\begin{equation}\label{notation10}
    H^r_\sharp= W^{r,2}_\sharp= H^r(\Gamma)= W^{r,2}(\Gamma).
\end{equation}

\begin{remark}\label{temporal} In Sections \ref{results}, \ref{appendixb}, and
\ref{section7}, we will consider one-parameter families of curves
$\Gamma(t)$, $t\in (0,T)$, with the lengths uniformly bounded from
above and uniformly separated from $0$. In this case the Sobolev
spaces of periodic functions depend on the temporal variable $t$ and
should be denoted by $H^r_\sharp(t)$. By abuse of notation, further
we omit the symbol $t$ and will write $H^r_\sharp$ instead of
$H^r_{\sharp}(t)$.
\end{remark}





\paragraph{Inequalities} Further,  we will use the simplest one-dimensional versions of the
Sobolev, interpolation, and Gagliardo-Nirenberg inequalities. The
first is the embedding inequality
\begin{equation}\label{appb8}
    \|f\|_{L^\infty(0,\mathcal L)}\leq c \|
    f\|_{H^\sigma_\sharp}\text{~~for  all~~}\sigma>1/2;
\end{equation}
the second is the standard interpolation inequality
\begin{equation}\label{appb9}
    \|f\|_{H^\varrho_\sharp}\leq c\,\,  \|f\|_{L^2(0,\mathcal L)}^{1-\frac{\varrho}{r}}\,\,\, \|
    f\|_{H^r_\sharp}^{\frac{\varrho}{r}}\text{~~for  all~~}0\leq\varrho\leq
    r;
\end{equation}
and the third is the Gagliardo-Nirenberg inequality, \cite{nir},
\begin{equation}\label{appb10}
    \|\partial_s^\varrho\,f\|_{L^{\frac{2r}{\varrho}}(0,\mathcal L)}\leq
    c\,\,
     \|f\|_{L^\infty(0,\mathcal L)}^{1-\frac{\varrho}{r}}\,\, \,\|
    f\|_{H^r_\sharp}^{\frac{\varrho}{r}}\text{~~for  all~~}0<\varrho\leq r.
\end{equation}
We also will use the Moser inequality
\begin{equation}\label{appb11}
    \|u\,v\|_{H^r_\sharp}\leq c \|u\|_{L^\infty(0,\mathcal L)}\,\, \|v
    \|_{H^r_\sharp}+c \|v\|_{L^\infty(0,\mathcal L)}\,\, \|u
    \|_{H^r_\sharp}, \quad 0\leq r<\infty.
\end{equation}
Here the constant $c$ depends only on $\mathcal L$ and the exponents
$\sigma$, $\varrho$, $r$.
\paragraph{Sobolev spaces on real line.}
 For every integer $r\geq
0$, denote by $H^r(\mathbb R)$, the Sobolev  space of mappings
$f:\mathbb R\to\mathbb R$ with the finite norm
\begin{equation}\label{appb7re}
    \|f\|^2_{H^r(\mathbb R)}=\int_{\mathbb R} (|f|^2+|\partial_s^r f|^2)\, ds.
\end{equation}
For real $r\geq 0$,  the norm in $H^r(\mathbb R)$ may be defined by
the equality
\begin{equation*}
 \|f\|^2_{H^r(\mathbb R)}=\int_{\mathbb R}(1+|\xi|^{2})^r |\hat{f}(\xi)|^2\,d\xi,
\end{equation*}
where the Fourier transform
$$
\hat{f}(\xi)=\frac{1}{\sqrt{2\pi}}\int_{\mathbb R} e^{-s\xi\, i}\,\,
f(s)\, ds.
$$
We also will use the denotation
$$
H^r(\mathbb R)= W^{r,2}(\mathbb R).
$$



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%% SECTION 3 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Results}\label{results}
\renewcommand{\theequation}{3.\arabic{equation}}
\setcounter{equation}{0}

\subsection{Estimates of Kohn-Vogelius functional}\label{vogelius}

The existence   of smooth solutions to the gradient flows equations
for  shape optimization problems guarantees that the steepest
descent method is well defined and gives the robust algorithm for
numeric calculations of an optimal shape. In this section we give
outline of main ideas of the proofs of existence and smoothness
results for the gradient flows in the shape optimization theory. In
order to be clear, we restrict our considerations to the    single
measurement identification problem for the Kohn-Vogelius functional.
Recall the formulation of this problem given in Section
\ref{preface}. Let us consider a simply connected bounded {\it hold
all} domain $\Omega\subset \mathbb R^2$ with an
 inclusion  $\Omega_i\subset\Omega$ bounded by a Jordan curve $\Gamma$. The
 interface $\Gamma$ splits $\Omega$ into the simply connected inclusion $\Omega_i$
 and two-connected curvilinear annulus $\Omega_e=\Omega\setminus
 \overline{\Omega}_i$. Define the conductivity coefficient $a$ by the relations
\begin{equation}\label{grad2}
    a=1\text{~~in~~}\Omega_e,  \quad
    a=a_0=\text{const.}>0\text{~~in~~}\Omega_i.
\end{equation}
Finally, choose  arbitrary functions $g, h:\partial\Omega\to\mathbb
R$ satisfying the conditions
\begin{equation}\label{drag3}
    h\in W^{1/2,2}(\partial\Omega), \quad g\in L^2(\Omega), \quad
    \int_{\partial \Omega} g\,ds=0.
\end{equation}
The Kohn-Vogelius energy functional  is defined as follows,
\cite{kohnvog},
\begin{equation}\label{shopt2222}
    J(\Omega_i)=\int_\Omega a\nabla(v-w)\cdot\nabla(v-w)\,dx.
\end{equation}
Here   $v,w:\Omega\to \mathbb R$ satisfy the equations and boundary
conditions
\begin{align}
%\notag
\text{div~}a\nabla v  & = 0 & \text{div~}a\nabla w  & = 0 &
\text{in~~} & \Omega, \label{grad4prim}
\\   \label{grad4}
a\nabla v\cdot  n & = g & w & = h & \text{on~~} & \partial\Omega,\\
\label{grad4bis} \int_{\partial\Omega} v \,dx&=0.&
\end{align}
Under the above assumptions, boundary value problems
\eqref{grad4prim}-\eqref{grad4} admit the only weak solutions
$v,w\in W^{1,2}(\Omega)$ satisfying the orthogonality condition
\eqref{grad4bis}, such that
\begin{equation}\label{grad5}
    \|v\|_{W^{1,2}(\Omega)}\leq c\|g\|_{L^2(\partial\Omega)}, \quad
\|w\|_{W^{1,2}(\Omega)}\leq c\|h\|_{W^{1/2,2}(\partial\Omega)}.
\end{equation}
Here $c$ depends only on $\Omega$ and the constant $a_0$ in the
definition \eqref{grad2} of $a$. Hence the Kohn-Vogelius functional
is well defined as a function of $\Omega_i$ or equivalently of
$\Gamma$.

 Assume, in addition, that the data have additional smoothness
properties
\begin{equation}\label{grad6}
\partial\Omega, \Gamma\in C^{2+\alpha}, \quad h\in C^{2+\alpha}(\partial\Omega),
\quad g\in C^{1+\alpha}(\partial\Omega)), \quad \alpha\in (0,1).
\end{equation}
Denote by $v^-,w^-$ the restrictions of $v$, $w$ on $\Omega_e$ and
by $v^+,w^+$ the restrictions of $v$, $w$ on $\Omega_i$. It follows
from the Schauder estimates for solutions to elliptic equations that
$v^-, w^-\in C^{2+\alpha}(\overline{\Omega}_e)$ and $v^+, w^+\in
C^{2+\alpha}(\overline{\Omega}_i)$. For every function $\Phi$ with
$\Phi^-$ and $\Phi^+$ continuous in $\overline{\Omega}_e$ and
$\overline{\Omega}_i$, the denotation $\big[\Phi\big]$, stands for
the jump of $\Phi$ across $\Gamma$,
$$
\big[\Phi\big](x)= \lim\limits_{\Omega_e \ni y \to x}\Phi^-
(y)-\lim\limits_{\Omega_i \ni y \to x}\Phi^+ (y)\text{~~for all~}
x\in\Gamma.
$$
For strong solutions to transmission problem \eqref{grad4} we have
\begin{equation}\label{grad7}
\big[a\partial_n v\big]\equiv \big[a\nabla v\big]\cdot n=0, \quad
\big[a\partial_n w\big]\equiv \big[a\nabla w\big]\cdot n=0, \quad
\big[v\big]=\big[w\big]=0.
\end{equation}
With this notation the gradient $dJ$ of the Kohn-Vogelius objective
function \eqref{shopt2} is defined as follows, see \cite{ADK},
\begin{equation}\label{grad8}
d J= 2\big(a\partial_n v\,\big[\partial_n v\big]- a\partial_n
w\,\big[\partial_n w\big])\, n-\big[a\nabla v\cdot\nabla v-a\nabla
w\cdot\nabla w\big]\, n,
\end{equation}


\subsubsection{Estimates of $dJ$}

In this section  we consider in details the gradient $dJ$ of the
Kohn-Vogelius functional.  Our goal is to derive the  estimates of
$dJ$ in the Sobolev spaces $H^r_\sharp$  in terms of the geometric
characteristics of the interface $\Gamma$. By virtue of
representation \eqref{grad8}, the normal vector field
$dJ:\Gamma\to\mathbb R^2$ is the quadratic form of the derivatives
of solutions $v$, $w$ to boundary value problems
\eqref{grad4prim}-\eqref{grad4bis}. First we derive the estimates
for a general transmission problem. Assume that the interface
$\Gamma$ satisfies the following conditions
\begin{itemize}
\item[\textbf{H.1}] The Jordan curve $\Gamma\subset \Omega$ satisfies the
energy condition
$$
\frac{1}{2}\int_\Gamma k^2ds +\mathcal L\leq E_0.
$$

\item[\textbf{H.2}] There
is $\nu>0$ with the property
\begin{equation*}
    \text{\rm dist~}(\Gamma\setminus \Gamma_{3\kappa}, \,
    \Gamma_{2\kappa})\,\geq \nu,
\end{equation*}
for every arc $\,\Gamma_{3\kappa}$ with  $\kappa$,  defined by Lemma
\ref{bokr3a}.


\item[\textbf{H.3}] There is $\varrho>0$ such that $\text{dist~}(\Gamma,
\partial \Omega)>\varrho$.
\end{itemize}
By virtue of Corollary \ref{bokr3st}, every curve $\Gamma$
satisfying Conditions \textbf{H.1}- \textbf{H.3} is a Jordan curve
of the class $C^{1+\alpha}$, $0<\alpha<1/2$. It splits the domain
$\Omega$ into two parts. The first $\Omega_i\Subset\Omega$
(inclusion) is a simply connected domain with boundary $\Gamma$. The
second is the curvilinear annulus $\Omega_e=\Omega\setminus
\overline{\Omega_i}$ bounded by $\Gamma$ and $\partial\Omega$. For
simplicity, we will assume that $\partial\Omega$ is a Jordan curve
of the class $C^\infty$. We adopt the convention that $\Gamma$ has
the positive orientation. This means that the point $z(s)$ moves
along $\Gamma$ in the counter-clockwise direction while $s$
increases. In its turn, the tangent vector $\tau$ and the normal
vector $n$ form the moving orthonormal frame with the positive
orientation. This means that $n$ is inward normal vector to
$\partial \Omega_i=\Gamma$.


Next, introduce the piece-wise constant function $a:\Omega\to
\mathbb R^+$ (conductivity) defined by the equalities \eqref{grad2}.

\paragraph{Model transmission problem.} Let $w\in W^{1,2}(\Omega)$ be a weak
 solution to the equation
\begin{equation*}
    \text{~div~}(a\,\nabla w)=0\text{~~in~~} \Omega.
\end{equation*}
We do not impose boundary conditions on $w$. Denote by $w^-$ and
$w^+$ the restrictions of $w$ onto subdomains $\Omega_e$ and
$\Omega_i$,
\begin{equation*}
w^-:=w\text{~~in~~}\Omega_e, \quad w^+:=w\text{~~in~~}\Omega_i.
\end{equation*}
If $\Gamma$ is sufficiently smooth, then $w$ is continuous on
$\Gamma$. In other words, $w^-=w^+$ on $\Gamma$. However, the normal
derivative of $w$ has a jump across $\Gamma$.  Next set
\begin{equation*}
\partial_n w^-=\nabla w^-\cdot n, \quad \partial_n w^+=\nabla
w^+\cdot n\text{~~on~~} \Gamma.
\end{equation*}
Our task is to estimate  $\partial_n w^{\pm}$
 via the curvature of $\Gamma$.
The following theorem on the estimates of $\partial_n w^\pm$  is the
first main result of this section. Recall definition
\eqref{notation10} of the  Sobolev spaces $H^r_\sharp=H^r(\Gamma)$
of periodic functions.

\begin{theorem}\label{normder1}
Under the above assumptions, the estimate
\begin{equation}\label{esty6}
    \|\partial_n w^\pm\|_{H^{m+1/2}_\sharp}\leq c\,(\,1+\|
    \partial_s^m
    k\|_{L^2(\Gamma)}\,)\,\|w\|_{W^{1,2}(\Omega)}
    \end{equation}
 holds for every integer $m\geq 0$. Here $c$ depends only on $m$ and
on the constants $E_0$, $\nu$, $\rho$ in Conditions {\rm
\textbf{H.1}-\textbf{H.3}}.
\end{theorem}
\textbf{Proof.} The proof is given in Section \ref{model}. $\square$


 \paragraph{Estimates of $dJ$.} Note that the  solutions $v,w$ to problems
 \eqref{grad4prim}-\eqref{grad4bis} meet al
requirements of Theorem \ref{normder1} and admit the estimates
\begin{equation*}
    \|v\|_{W^{1,2}(\Omega)}+ \|w\|_{W^{1,2}(\Omega)}\leq c(g,h).
\end{equation*}
This result along with representation \eqref{grad8} and the
multiplicative estimates in Sobolev spaces leads to the following
theorem, which is the second main result of this section.






\begin{theorem}\label{bokr60}
Assume that a curve $\Gamma$ satisfies conditions {\rm
\textbf{H.1}-\textbf{H.3}} and $k\in H^r_\sharp$, for some integer
$r\geq 1$.
 Then for every $\beta\in [0,1/2)$, there is a constant $c$,
depending on $r$, $\beta$, and constants $E_0$, $\nu$, $\rho$ in
conditions {\rm \textbf{H.1}-\textbf{H.3}}, such that the gradient
$dJ(s)$ of the Kohn-Vogelius functional admits the estimate
\begin{equation}\label{bokr61aa}
    \| dJ\|_{H^{r+\beta}}\leq c(1+\|k\|_{H^r_\sharp}).
\end{equation}
In particular, we have
\begin{equation}\label{bokr61b}
    \|\partial_s^r dJ\|_{L^q(0,\mathcal L)}\leq c(1+\|k\|_{H^r_\sharp}).
\end{equation}
for every $q\in [1,\infty)$. In this case the constant $c$ depends
in addition on $q$.
\end{theorem}






\subsection{Gradient flow. Existence theory}


\subsubsection{Problem formulation} The standard formulation of the geometric
flow equations deals with immersions (parametrized curves). Further
we will assume that the
 interface $\Gamma$ admits the representation $\Gamma=f(\mathbb S^1)$, where
the immersion  $f: \mathbb S^1\to \mathbb R^2$ is unknown and should
be defined along with the solution to the gradient  flow problem
\eqref{gala14}.  Note that $f$ is a $2\pi$ periodic function of the
angle variable $\theta\in \mathbb R/2\pi\mathbb Z$. The element of
the length of $\Gamma$ equals
$$ds=\sqrt{ g(\theta)}\, d\theta,\quad
g=|\partial_\theta f|^2,
$$
where $g$ is the only nontrivial coefficient of the first
fundamental form of the curve $\Gamma$. In this setting, the
derivative with respect to the arc-length variable $s$
$$
\partial_s=\frac{1}{\sqrt{g}(\theta)}\partial_\theta
$$
becomes the nonlinear differential operator depending on $f$.

  The tangent vector
$$\tau(\theta)=\partial_sf(\theta):=|\partial_\theta
f|^{-1}\, \partial_\theta f(\theta),
$$
and the normal vector
$$
 n(\theta)=\tau^\bot(\theta)=(-\tau_2,\, \tau_1),
$$
form the positive oriented moving frame on $\Gamma$. Notice that $
n$ is the unit inward normal vector to $\partial \Omega_i=\Gamma$.
The  curvature vector $ k$ is defined by the equalities
\begin{equation}\label{gala18}\begin{split}
 k(\theta)=\partial_s \tau(\theta)=\partial_s^2f(\theta).
\end{split}\end{equation}
 Notice that
$ k$ is orthogonal to $\tau$ and is directed along the normal vector
$ n$.

The Euler elastic energy $\mathcal E_e$ and the perimeter $\mathcal
L$ are defined by the equalities
\begin{equation}\label{grad9}
    \mathcal E_e=\int_{\Gamma}
    \frac{k^2}{2}\, ds, \quad \mathcal L=\int_{\Gamma}
\, ds=\int_0^{2\pi}\sqrt{g}\, d\theta.
\end{equation}
 We  take the penalization energy in the form
\begin{equation}\label{gala16}
 \mathcal E=\mathcal E_e+\mathcal L=\int_{\Gamma}\big(\,
    \frac{k^2}{2}+1\big)\, ds,
\end{equation}
 The gradient of
$\mathcal E$ is given by the following lemma.


\begin{lemma}\label{eulervar3}
Under the above assumptions, we have
\begin{equation}\label{eulervar4}
d \mathcal E_e(f)= \nabla_s\nabla_s\, k+\frac{1}{2}| k|^2\, k, \quad
d\mathcal L=- k,
\end{equation}
\begin{equation}\label{gala17}
d \mathcal E(f)= \nabla_s\nabla_s\, k+\frac{1}{2}| k|^2\, k- k.
\end{equation}
Here the normal connection $\nabla_s$ for every vector field
$\Phi:\Gamma\to \mathbb R^2$, is defined by the equality
\begin{equation}\label{gala18a}
\nabla_s\, \Phi=\partial_s \Phi-(\partial_s \Phi\cdot\tau)\, \tau.
\end{equation}
\end{lemma}
Identities  \eqref{gala17} are classic (see for instance
\cite{DKS}). They are very particular case of the $3D$ Willmore
variation formula.



 We are now in a position to specify the gradient flow equation
\begin{equation*}
    \partial_t f+d\mathcal E+d J=0, \quad f(0)= f_0.
\end{equation*}
for the penalized Kohn-Vogelius functional.  Applying Lemma
\ref{eulervar3} we can rewrite this equation in the form
\begin{equation}\label{gala19}
    \partial_t f+\,\nabla_s\nabla_s\,
k+\frac{1}{2}| k|^2\, k- k+d J=0\text{~for~} t>0, \,\,\, f(0)=f_0.
\end{equation}
The gradient $dJ$ is defined by relation \eqref{grad8} and can be
regarded as nonlinear nonlocal operator acting on $\Gamma$. Hence
\eqref{gala19} is a nonlinear operator equation. It may be
considered as a nonlocal perturbation of the Euler elastic flow.







\subsubsection{Existence theorem}

 In this subsection we prove the main
theorem on the existence of global smooth solution to problem
\eqref{gala19}. Assume that the initial data satisfy the following
conditions:
\begin{itemize}
\item[
{\bf I.1}] The even integer number $m\geq 10$

\item[{\bf I.2}] The initial curve $\Gamma_0= f_0(\mathbb S^1)$
 satisfies conditions {\bf H.1}-{\bf H.3} of Theorem \ref{normder1}.

\item[{\bf I.3}]There is a constant $E_m$ such that
\begin{equation}\label{exist2a}
    \int_{\Gamma_0}|\partial_s^{r} k_0|^2\, ds \leq E_m\text{~~for
    all~~} 0\leq r\leq m.
\end{equation}
\item[{\bf I.4}] The length element $\sqrt{g_0}=|\partial_\theta
f_0|$ satisfies the condition
\begin{equation}\label{exist2}
    \|\sqrt{g_0}\|_{C^{m-5}(\mathbb S^1)}\leq c_g<\infty.
\end{equation}
\end{itemize}



\begin{theorem}\label{theoremexist} Assume that the initial data
satisfy Conditions {\bf I.1}-{\bf I.4}. Then there is a maximal
$T\in (0,\infty]$ with the following properties. Problem
\eqref{gala19} has a solution  such that
\begin{equation}\label{asol24}
f\in C(0,T'; C^{m-5}(\mathbb S^1)), \,
    \partial_t f\in C(0,T'; C^{m-9}(\mathbb S^1)) \text{~for all~} 0<T'<T.
\end{equation}
Moreover, the Jordan curves $\Gamma(t)= f(t, \mathbb S^1)$, $t\in
[0,T)$, are separated from $\partial\Omega$ and have no
self-intersections. If~ $T<\infty$, then there is a sequence
$f(t_j)$, $t_j\to T$ as $j\to\infty$, such that $\text{\rm
~dist~}(\Gamma(t_j),
\partial\Omega)\to 0$ , or  (and) $f(t_j)$ converge in
$C^1(\mathbb S^1)$ as $j\to \infty$ to some immersion $f_\infty$
such that the curve $f_\infty(\mathbb S^1)$ has a self-intersection.
\end{theorem}

 The proof is standard and consists of three steps. The first is
the  proof of the local solvability of problem \eqref{gala19} on the
small time intervals. The second most important step is the proof of
the global a priori estimates for smooth solutions to problem
\eqref{gala19} in Sobolev and H\"{o}lder classes. These estimates
and the extension method  entail the existence of  smooth solution
which meets all requirements of Theorem  \ref{theoremexist}.

A detailed proof of short-time existence is outside of the scope of
this paper.  Note that equation \eqref{gala19} is a degenerate
parabolic equation with added lower order operator $dJ$.  In our
case the local existence result  can be obtained by writing the flow
as a graph over the initial date. In particular, the problem can be
reduced to a scalar parabolic equation for the distance function,
\cite{chou}. See also \cite{DKS} and \cite{K} for useful arguments.

Hence out main task is to derive global a priori estimates for
solutions to problem \eqref{gala19}. This part of the proof is
technical and lengthy. Our approach  is  based on the estimates for
Kohn-Vogelius functional given by   Theorem \ref{bokr60} and methods
developed in \cite{AP}, \cite{DKS}, and \cite{L}. The results are
given by the following two theorems. The first constitutes the
Sobolev a priori estimates for the curvature $k$ as a function of
the arc-length variable $s$.

\begin{theorem}\label{theoremsobolev} Let $f:[0,T]\times \mathbb S^1\to
\mathbb R^2$ be a smooth solution to problem \eqref{gala19} with
initial data satisfying condition
\begin{equation}\label{apriori1}
\int_{\Gamma(0)}|\partial_s^m k_0|^2 ds\leq E_m<\infty,\quad
\mathcal E(0)\leq E_0<\infty,
\end{equation}
where $m\geq 6$ is an even integer. Furthermore assume that there
are two positive constants $\nu$ and $\rho$ with the following
properties.  For every $t\in [0,T]$, the curve $\Gamma(t)$ satisfies
Conditions {\rm \textbf{H.1}-\textbf{H.3}} with  fixed constant
$\nu$ and $\text{\rm dist~}(\Gamma(t),
\partial \Omega)>\varrho$ independent of $t$. Then
there is a  constant $c$, depending only on $E_0$, $\nu$, $\rho$,
$T$, and  $m$, such that
\begin{equation}\label{appb6}
\sup_{t\in [0,T]}\|k(t)\|_{H^{m-2}(t)}^{2} +\int_0^T\|
k(t)\|_{H^m(t)}^2\, dt\leq c E_m+c(1+T).
\end{equation}
\end{theorem}
\begin{proof} The proof is given in Section \ref{appendixb}.
\end{proof}


The second theorem gives the  a priori estimates for solutions to
problem \eqref{gala19} in the H\"{o}lder classes.
\begin{theorem}\label{theoremsmooth}
Let a smooth solution to problem \eqref{gala19} meets all
requirements of Theorem \ref{theoremsobolev} with even integer
$m\geq 10$. Furthermore assume that the initial data satisfies
conditions {\bf I.1}-{\bf I.4} of Theorem \ref{theoremexist}. Then
there is a constant $c$, depending only on $T$, $\nu$, $\varrho$,
$m$ and the constants $E_m$, $c_g$ in conditions {\bf I.1}-{\bf
I.4}, such that
\begin{equation}\label{appb6sm}
\|f\|_{C(0,T; C^{m-5}(\mathbb S^1))} +\| f\|_{C^1(0,T;
C^{m-9}(\mathbb S^1))}\leq c.
\end{equation}
\end{theorem}
\begin{proof} The proof is given in Section \ref{section7}.
\end{proof}


In order to complete the proof of Theorem \ref{theoremexist} we use
the extension method. Without loss of generality we may assume that
$f_0\in C^\infty(\mathbb S^1)$. Hence the problem has a $C^\infty$-
solution $f$ defined on some small interval $(0,T)$.  By virtue of
Theorem \ref{theoremsmooth}, this solution meets all requirements of
Theorem \ref{theoremexist}~for every even  $m<\infty$. Moreover,
every immersion $f(t)$, $t\in [0,T)$,   satisfies conditions
\textbf{H.1}-\textbf{H.3} of Theorem  \eqref{bokr60} with some
constants  $\nu(t)>0$ and $\varrho(t)>0$. Let $(0,T)$ be the maximal
interval of existence of such a solution. There are two
possibilities
\begin{equation*}
\text{either~~}\liminf\limits_{t\to
T}\nu(t)\varrho(t)>0\text{~~or~~}\liminf\limits_{t\to
T}\nu(t)\varrho(t)=0.
\end{equation*}
Let us prove that $T=\infty$ in the first case.  Assume in contrary
to our claim that $T<\infty$. There is $\delta>0$ such that
quantities $\nu(t)$ and $\varrho(t)$ are uniformly separated from
zero on the interval $[T-\delta,T)$, i.e.,
$$
\nu(t)>\nu>0\quad \varrho(t)>\rho>0
$$
for some $\nu$ and $\rho$. Hence $f(t)$ meet all requirements of
Theorem \ref{theoremsmooth} on the interval $[T-\delta,T)$ with the
initial data $f(T-\delta)$.  It follows from this theorem that
$$
\|f(t)\|_{C^{m-5}(\mathbb S^1)}+\|\partial_t f(t)\|_{C^{m-9}(\mathbb
S^1)} \leq c(m)\text{~~for all~~} t\in[T-\Delta,T].
$$
Recall that here $m\geq 10$ is an arbitrary even integer. Hence the
immersions $f(t)$ converges in every space $C^m(\mathbb S^1)$ to
some immersion $f_\infty\in C^\infty(\mathbb S^1)$ which obviously
satisfies conditions  {\bf I.1}-{\bf I.4}. The local existence
theory implies the existence of smooth solution to equation
\eqref{gala19} with initial data $f(T)=f_\infty$ on some interval
$[T,T+\delta)$. This contradicts the maximality of $T$.

It remains to consider the case when $T<\infty$ and hence
$\liminf\nu(t)\varrho(t)=0$. Obviously there exist a sequence $t_j$
such that
$$
\nu(t_j)\rho(t_j)\to 0, \quad t_j\to T\text{~~as~~}j\to \infty.
$$
If $\varrho(t_j)\to 0$ as $t_j\to T$, then
$\text{dist~}(\Gamma(t_j),\partial \Omega)\to 0$ as $t_j\to T$ and
the assertion follows. Let us consider the case
\begin{equation}\label{immer100}
\nu(t_j)\to 0, \quad t_j\to T\text{~~as~~}j\to \infty.
\end{equation}
Recall that the immersions $x= f(t, s)$, $s\in [0, \mathcal L(t)]$,
are uniformly bounded in $C^{1+\alpha}[0, \mathcal L(t)]$. Moreover,
the bound depends only on the constant $E_0$. Furthermore, by virtue
of Lemma \ref{bokr2}, the perimeters $\mathcal{L}(t)$ are uniformly
bounded from above and uniformly separated from $0$.  After passing
to a subsequence we may assume that the sequence $\mathcal L(t_j)$
converges to some positive $\mathcal L_\infty$ as $t_j\to T$. The
sequence of immersions $f(t_j,s)$ converges in $C^1$ norm  to an
immersion $f_\infty(s)$ on every compact subset of $[0, \mathcal
L_\infty)$. It is clear that the energy of the correspondent curve
$\Gamma_\infty$ does not exceed $E_0$.  It remains to prove that the
limiting curve $\Gamma_\infty$ has a self-intersection. To this end,
note that the set of curves $\{\Gamma(t_j)\}\cup \Gamma\infty$ is
compact in the uniform metric. If the limiting curve has no self-
intersections, then every curve from this set has
no-self-intersections. From this and Corollary \ref{immer2g} we
conclude that $\nu(t_j)\geq \nu>0$ for some $\nu$ independent of
$j$, which contradicts to relation \eqref{immer100}. This completes
the proof of Theorem \ref{theoremexist}.



Since the energy $\mathcal E(t_j)$ of the curve $\Gamma(t_j)$ is
bounded by the constant $E_0$, it follows from Lemma \ref{bokr3}
that the functions $f_j(s)=f(t_j,s)$ are uniformly bounded in
$C^{1+\alpha}$ norm for $0\leq\alpha<1/2$. Hence after passing to a
subsequence we may assume that $\Gamma(t_j)$ convege uniformly to
$C^1$ curve $\Gamma_\infty$. Obviously either $\Gamma_\infty$ has a
self-intersection or (and) it touches $\partial\Omega$.  This
completes the proof of Theorem \ref{theoremexist}.



















%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%% SECTION 4 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%











 \section{Model transmission problem}\label{model}
\renewcommand{\theequation}{4.\arabic{equation}}
\setcounter{equation}{0}
\subsection{Transmission problem. Notation.
Results.}
 Let us consider the following construction. Fix an arbitrary
positive $\kappa$ an $\rho$ and introduce the rectangles
\begin{equation}\label{tran1}
Q_0=(-2\kappa, 2\kappa)\times (-2\rho,2\rho),\quad Q=(-\kappa,
\kappa)\times (-\rho,\rho)
\end{equation}
in the plane of variable $y=(y_1,y_2)$.   Next, fix an arbitrary
integer $r\geq 1$ and introduce the systems of numbers
$$
\kappa_m=\kappa\big(\,2-\frac{m}{r}\,\big), \quad
\rho_m=\rho(2-\frac{m}{r}), \quad 1\leq m\leq r,
$$
and the corresponding domains
\begin{equation}\label{tran2}
Q_m=(-\kappa_m,\kappa_m)\times (-\rho_m,\rho_m),\quad Q_r=Q.
\end{equation}
Choose an arbitrary function $\varphi$ with the properties
\begin{equation}\label{tran3}
\varphi\in C^\infty_0(Q), \quad 0\leq \varphi\leq 1, \quad
\varphi=1\text{~~in~~} Q/2, \quad
\partial_{y_2}
\varphi=0\text{~for~} y_2=0.
\end{equation}
and a system of functions $\varphi_m$, $1\leq m\leq r$,  such that
\begin{equation}\label{tran4}\begin{split}
\varphi_m\in C^\infty_0(Q_m), \quad 0\leq \varphi_m\leq r-1, \\
\varphi_m=1\text{~~in~~} Q_{m+1} \text{~~for~~}1\leq m\leq r-1,
\quad \varphi_r=\varphi. \end{split}\end{equation} Next, introduce
($2\times 2$)-matrix $N(y_1)$ with the properties
\begin{equation}\label{tran5}
N=N^\top, \quad  C_N^{-1}\mathbb I\leq N \leq C_N\mathbb I, \quad
\|N\|_{W^{1,2}(-2\kappa, 2\kappa)}\leq C_N,
\end{equation}
where $C_N$ is some fixed constant. Finally set
$$
a(y_2)=1\text{~~for~~}y_2<0, \quad
a(y_2)=a_0=\text{~const.}>0\text{~~for~~} y_2\geq 0.
$$

Let $u:Q_0\to \mathbb R$ be a solution to the elliptic equation
\begin{equation}\label{tran6}
    \text{~div~}(aN\nabla u)=0\text{~~in~~} Q_0.
\end{equation}
We do not impose  boundary conditions for $u$. Instead of this we
assume that it admits the estimate
\begin{equation}\label{tran7}
    \|u\|_{L^2(Q_0)}+\|\nabla u\|_{L^2(Q_0)}\leq C_u<\infty
\end{equation}
It follows from $L^\infty$ interior estimates for solutions of
divergent elliptic equations that
\begin{equation}\label{tran7bis}
    \|u\|_{C(Q_m)}\leq c<\infty\text{~~for~~}m\geq 1,
\end{equation}
where the constant $c$ depends only on $C_N$, $C_u$, $a$, and
$\kappa$, $\rho$.


Now set
\begin{equation}\label{tran8}
v=\varphi \,u,
\end{equation}
where $\varphi$ is defined by \eqref{tran3}.


\emph{The main goal of this subsection is to estimate the one-sided
conormal derivatives of $v$ on the interface $\{x_2=0\}$.} To this
end we introduce the system of functions
\begin{equation}\label{tran9a}
v_m=\varphi_m\, u,\quad 1\leq m\leq r,
\end{equation}
where $\varphi_m$ are given by \eqref{tran4}.


\begin{proposition}\label{tran59}
Under the above assumptions,  the functions $v_m=\varphi_m u$,
$1\leq m\leq r$, admit the estimates
\begin{equation}\label{tran60}
    \|\partial_1^m \nabla v_m\|_{L^2(Q_m)}\leq c\,(\,1+\|\partial_1^m
    N\|_{L^2(-2\kappa, 2\kappa)}\,).
\end{equation}
 Here the constant $c$ depends only on $r$, $\varphi$, and the
constants $C_N$, $C_u$ in \eqref{tran5} and \eqref{tran7}.
\end{proposition}

The second main result of this section is the following proposition,
which provides the estimates for the conormal derivatives of $u$ on
the interface $\{y_2=0\}.$ Split the rectangle $Q$ into two parts
$$
Q^-=(-\kappa, \kappa)\times (-\rho,0), \quad Q^+=(-\kappa,
\kappa)\times (0,\rho),
$$
separated by the interface segment
$$
\ell= (-\kappa, \kappa)\times \{0\}.
$$
Denote by $u^\pm$ and $v^\pm$ the restrictions of the functions $u$
and $v=\varphi u$ to $Q^{\pm}$. We also denote  by $\partial_N
u^\pm$ and $\partial_N v^\pm$  the conormal derivatives
\begin{equation}\label{tran70}
\partial_N u^\pm=(N_{21}\partial_1+N_{22}\partial_2)\,u^\pm,\quad
\partial_N v^\pm=(N_{21}\partial_1+N_{22}\partial_2)
v^\pm\text{~~on~~}
%\mathbf l
\ell.
\end{equation}
We will consider the conormal derivatives as functions defined on
the interval $(-\kappa,\kappa)$. Since the function $\varphi$ is
compactly supported in this interval, we may assume that
$\varphi\partial_N u^\pm$ and $\partial_N v^\pm$ are extended by
zero to $\mathbb R$. Now recall definition of Sobolev spaces
$H^r(\mathbb R)$ in subsection \ref{sobolevfun}.

\begin{proposition}\label{tran72}
Assume that all assumptions of Proposition \ref{tran59} are
satisfied. Then the estimate
\begin{equation}\label{tran73}
    \|\varphi\partial_N u^{\pm}\|_{W^{r-1/2,2}(\mathbb R)}\leq
    c(1+\|\partial_1^r N\|_{L^2(-2\kappa, 2\kappa)})
\end{equation}
holds true for all integers $r\geq 1$.

\end{proposition}
The rest of the subsection is devoted to the proof of Propositions
\ref{tran59} and \ref{tran72}.

\subsection{Proof of Proposition \ref{tran59}. Part I} We proceed with the
induction principle estimating step by step the function $v_m$. Our
first step is the derivation of  recurrent system of elliptic
equations for these $v_m$ and their derivatives.

\subsubsection{Extended system of equations.}
 It is easily seen that the functions $ v_m$, $1\leq m\leq r$ satisfy the following
recurrent system of differential equations
\begin{equation}\label{tran9}
\text{div~}\big(a N\nabla v_m)=\text{~div~}\mathcal B_{m-1}+\mathcal
C_{m-1}\text{~~in~~} Q_0,\quad 1\leq m\leq r,
\end{equation}
which is understood in the sense of distributions. Here $v_0=u$ and
\begin{equation}\label{tran10}
\mathcal B_{m-1}=v_{m-1}\,a\, N\nabla\varphi_m, \quad \mathcal
C_{m-1}=a\nabla\varphi_m\cdot N\,\nabla v_{m-1}.
\end{equation}
Indeed, we have
\begin{gather*}
\int_{Q_0} aN\nabla v_m\cdot \nabla\zeta\, dy=\int_{Q_0}
aN\nabla(\varphi_m u)\cdot \nabla\zeta\, dy=\\
\int_{Q_0} a N v_{m-1}\nabla\varphi_m \cdot \nabla\zeta\, dy +
\int_{Q_0} a N \varphi_m \nabla u\cdot \nabla\zeta\, dy.
\end{gather*}
On the other hand, we have
\begin{gather*}
\int_{Q_0} a N \varphi_m \nabla u\cdot \nabla\zeta\, dy= \int_{Q_0}
a N \nabla u\cdot\nabla(\zeta\,\varphi_m) dy-\int_{Q_0} a N\nabla
u\cdot\nabla\varphi_m \,\zeta\,
dy\\
=-\int_{Q_0} a N\nabla u\cdot\varphi_m \,\zeta\, dy=-\int_{Q_0} a
N\nabla v_{m-1}\cdot\nabla\varphi_m \,\zeta\, dy.
\end{gather*}
 Thus we get
$$
\int_{Q_0} aN\nabla v_m\cdot \nabla\zeta\, dy=\int_{Q_0} a
v_{m-1}N\nabla\varphi_m \cdot \nabla\zeta\, dy -\int_{Q_0} a N
\nabla\varphi_m \nabla v_{m-1}\zeta\,dy,
$$
which obviously yields \eqref{tran9}.  We are interested in the
smoothness properties of solutions to system
\eqref{tran9}-\eqref{tran10} with respect to the variable $y_1$. To
this end, notice that
\begin{equation}\label{tran11}
\partial_1^m\text{div~}\big(a N\nabla v_m)=\text{~div~}\partial_1^m\mathcal
B_{m-1}+\partial_1^m\mathcal C_{m-1}\text{~~in~~} Q_0,\quad 1\leq
m\leq r,
\end{equation}
and
\begin{equation}\label{tran12}
    \partial_1^m\text{div~}\big(a N\nabla v_m)=\text{div~}\big(a N\nabla\partial_1^m v_m)
    +\text{\rm~~div~}\mathcal A_m.
\end{equation}
Here
\begin{equation}\label{tran13}
    \mathcal A_m=\sum_{i+j=m, j\geq 1}\mathcal A_{ij}^m, \quad
    \mathcal A_{ij}^m= C^m_j\, a\partial_1^j N\, \nabla\partial_1^i v_m.
\end{equation}

Thus we get
\begin{equation}\label{tran14a}
\text{div~}\big(a N\nabla\partial_1^m v_m)=
    -\text{~div~}\mathcal A_m+\text{~div~}\partial_1^m\mathcal
B_{m-1}+\partial_1^m\mathcal C_{m-1}\text{~~in~~} Q_m
\end{equation}
for every $1\leq m\leq r$. Now our task is to estimate the
quantities $\mathcal A_m$, $\mathcal B_{m-1}$, $\mathcal C_{m-1}$,
and $\mathcal D_{m-1}$.






\subsubsection{Basis of induction. Auxiliary Lemma}
In this subsection we prove the following lemma, which gives the
basis of the induction process. In what follows, we will denote by
$c$ various constants depending on the rectangle $Q_0$, and $r$,
$\varphi_m$, as well as the constants $C_N$ and $C_u$ in
\eqref{tran5} and \eqref{tran7}.
\begin{lemma}\label{tran15}
Under the above assumptions,
\begin{equation}\label{tran16}
    \|\partial_1 \nabla v_1\|_{L^2(Q_1)}\leq c.
\end{equation}
\end{lemma}
\begin{proof} The proof is given in Section \ref{indbasis}.
\end{proof}
\begin{remark}\label{tran26}
Notice that
\begin{equation}\label{tran27}
 \|\partial_1\mathcal B_1\|_{L^2(Q_1)} +
 \|\mathcal C_1\|_{L^2(Q_1)}\leq c.
\end{equation}


\end{remark}


\paragraph{Auxiliary lemma.} Now we have to organize the induction process. To this end, we
have to derive the recursion  system of  estimates for  the
quantities $\mathcal A_m$, $\mathcal B_{m_1}$ and $\mathcal
C_{m-1}$. Notice that for $m=1$ the desired estimates   follows from
Remark \ref{tran26}. Hence it suffices to consider the case $m\geq
2$.
\begin{lemma}\label{tran28}
For every integer $m\in[2, r]$ and $\sigma\in (1/2,1)$, there is a
constant $c$ such that
\begin{equation}\label{tran29}
    \|\mathcal A_m\|_{L^2(Q_m)}\leq c(1+\|\partial_1^m
    N\|_{L^2(-\kappa_m,\kappa_m)}+\|\partial_1^m \nabla
    v_m\|_{L^2(Q_m)}^{\lambda}),
\end{equation}
where $\lambda=(m-2+\sigma)/(m-1)<1$,
\begin{equation}\label{tran41}\begin{split}
    \|\partial_1^{m-1}\mathcal C_{m-1}\|_{L^2(Q_m)}\leq\\
    c(\,1+\|\partial_1^{m-1}N\|_{L^2(-\kappa_m, \kappa_m)}+\|\partial_1^{m-1}\nabla
    v_{m-1}\|_{L^2(Q_m)}\, ),
\end{split}\end{equation}
\begin{equation}\label{tran58}\begin{split}
 \|\partial_1^m \mathcal B_{m-1}\|_{L^2(Q_m)}\leq\\
    c\big(1+\|\partial_1^m
    N\|_{L^2(-\kappa_m,\kappa_m)}+\|\partial_1^{m-1}\nabla
    v_{m-1}\|_{L^2(Q_{m-1})}\big).
\end{split}\end{equation}

\end{lemma}

\begin{proof}
The proof is given in Section \ref{lemma28}
\end{proof}






\subsection{Proof of Proposition \ref{tran59}. Part II} We are now in a
position to complete the proof of Proposition \ref{tran59}. Recall
the denotations for the rectangles $Q$ and $Q_0$,
$$
Q=(-\kappa,\kappa)\times (-\rho,\rho),\quad
Q_0=(-2\kappa,2\kappa)\times (-2\rho,2\rho).
$$
Let us consider the sequences of domains $Q=Q_r\subset
Q_{r-1}...\subset Q_0$ and functions $\varphi_m$, $0\leq m\leq r$,
defined by relations \eqref{tran2}-\eqref{tran4}. Recall that
$\varphi_r=\varphi$.  Let us also consider the sequence of the
functions $v_m=\varphi_m u$. It is necessary  to prove that the
estimate
\begin{equation}\label{tran63}
    \|\partial_1^m \nabla v_m\|_{L^2(Q)}\leq c(1+\|\partial_1^m
    N\|_{L^2(-\kappa, \kappa)})
\end{equation}
holds true for all $1\leq m\leq r$.  Notice that for $m=1$, this
estimate obviously follows from estimate  in Lemma \ref{tran15}. Now
we proceed with the induction principle. Assume that the inequality
\begin{equation}\label{tran65}
    \|\partial_1^{m-1} \nabla v_{m-1}\|_{L^2(Q_{m-1})}\leq
    c(1+\|\partial_1^{m-1}
    N\|_{L^2(-\kappa_{m-1}, \kappa_{m-1})})
\end{equation}
holds for some $m\geq 2$. Notice that the function $v_m$ satisfies
equation \eqref{tran14a} that reads
\begin{equation}\label{tran14z}
\text{div~}\big(a N\nabla\partial_1^m v_m)=
    -\text{~div~}\mathcal A_m+\text{~div~}\partial_1^m\mathcal
B_{m-1}+\partial_1^m\mathcal C_{m-1}\text{~~in~~} Q_m,
\end{equation}
where $\mathcal A_m$, $\mathcal B_{m-1}$, and $\mathcal C_{m-1}$ are
given by \eqref{tran10} and \eqref{tran13}. Recall that the positive
matrix $aN$ is uniformly bounded from below and above. Multiplying
both sides of \eqref{tran14z} by $\partial_1^m v_m$, integrating the
result by parts over $Q_m$, and applying the Cauchy inequality we
arrive at the estimate
\begin{equation*}
    \|\partial_1^m\nabla v_m\|_{L^2(Q_m)}\leq \|\mathcal
    A_m\|_{L^2(Q_m)}+\|\partial_1^m\mathcal B_{m-1}\|_{L^2(Q_m)}+
\|\partial_1^{m-1}\mathcal C_{m-1}\|_{L^2(Q_m)}.
\end{equation*}
It follows from estimates  in  Lemma \ref{tran28} that
\begin{equation*}\begin{split}
    \|\partial_1^m\nabla v_m\|_{L^2(Q_m)}\leq c +
 \|\partial_1^m\nabla v_m\|_{L^2(Q_m)}^\lambda+\\
c\|\partial_1^m N\|_{L^2(-\kappa_m, \kappa_m)}+c
 \|\partial_1^{m-1} \nabla v_{m-1}\|_{L^2(Q_{m-1})}),
\end{split}\end{equation*}
where $0<\lambda<1$. From this and the induction hypothesis we
conclude that
\begin{equation*}
    \|\partial_1^{m} \nabla v_m\|_{L^2(Q_{m})}\leq
    c\,\big(\,1+\|\partial_1^{m}
    N\|_{L^2(-\kappa_{m}, \kappa_{m})} \,\big).
\end{equation*}
This completes the proof of the induction step. Applying the
induction principle we obtain  desired estimate \eqref{tran60} for
all $m\in[1,r]$. This completes the proof of Proposition
\ref{tran59}.



\subsection{Conormal derivative. Proof of Proposition \ref{tran72}.}


Split the rectangle $Q$ into two parts
$$
Q^-=(-\kappa, \kappa)\times (-\rho,0), \quad Q^+=(-\kappa,
\kappa)\times (0,\rho),
$$
separated by the interface segment
$$
\ell= (-\kappa, \kappa)\times \{0\}.
$$
Denote by $u^\pm$ and $v^\pm$ the restrictions of the functions $u$
and $v=\varphi u$ to $Q^{\pm}$. We also denote  by $\partial_N
u^\pm$ and $\partial_N v^\pm$  the conormal derivatives
\begin{equation}\label{tran70con}
\partial_N u^\pm=(N_{21}\partial_1+N_{22}\partial_2)\,u^\pm,\quad
\partial_N v^\pm=(N_{21}\partial_1+N_{22}\partial_2)
v^\pm\text{~~on~~}\ell.
\end{equation}
We will consider the conormal derivatives as a function defined on
the interval $(-\kappa,\kappa)$. Since the function $\varphi$
vanishes for $|x_1|\geq \kappa$, we may assume that
$\varphi\partial_N u^\pm$ and $\partial_N v^\pm$ are extended by
zero to $\mathbb R$. Recall that for every $s\geq 0$, the Sobolev
space $W^{s,2}(\mathbb R)$ is defined as the completion of the space
$C^\infty_0(\mathbb R)$ with respect to the norm
\begin{equation}\label{tran71}
\|u\|_{W^{s,2}(\mathbb R)}^2=\int_{\mathbb R} (1+\xi^2)^s\,
|\hat{u}(\xi)|^2\,d\xi,
\end{equation}
where $\hat{u}$ is the Fourier transform of $u$.

 Now, we estimate  $\varphi\partial_N u^-$. The proof of inequality
\eqref{tran73} for $\varphi\partial_N u^+$ is similar. First we show
that such an inequality holds true for $\partial_N v^-$. It suffices
to prove  that the estimate
\begin{equation}\label{tran74}
\Big|\int_{\mathbb R}\partial_1^r(\partial_N v^-)\,\zeta\,
dy_1\Big|\leq c\, (1+\|\partial_1^r N\|_{L^2(-2\kappa, 2\kappa)}\,)
\|\zeta\|_{W^{1/2,2}(\mathbb R)}
\end{equation}
holds for all compactly supported $\zeta\in W^{1/2,2}(\mathbb R)$.
Recall that $v$ is extended by zero to $\mathbb R^2$. Choose an
arbitrary function $\zeta\in W^{1/2,2}(\mathbb R)$. By virtue of the
extension theorem for Sobolev functions, the function $\zeta$ can be
extended to the strip $Q_\infty=\mathbb R\times (-\rho,0)$ such that
the extension $\zeta^*$ admits the estimate
\begin{equation}\label{tran78}
    c^{-1}\|\zeta\|_{W^{1/2,2}(\mathbb R)}\leq
    \|\zeta^*\|_{W^{1,2}(Q_\infty)}\leq c \|\zeta\|_{W^{1/2,2}(\mathbb
    R)}.
\end{equation}
 Multiplying both the sides
of \eqref{tran11} by $\zeta$ and integrating the result by parts
over $Q^-_0$ we arrive at the integral identity
\begin{equation}\label{tran76}\begin{split}
    \int_{\mathbb R}\partial_1^r\partial_N v^-\, \zeta\, dy_1=
\mathbf I_1+\mathbf I_2+\mathbf I_3+\mathbf I_4,
\end{split}\end{equation}
where
\begin{equation}\label{tran76a}\begin{split}
\mathbf I_1=\int_{\mathbb R} \partial_1^r(v_{r-1}\partial_N
\varphi)\zeta\, dy_1,\\
\mathbf I_2=\int_{Q^-_0}\partial_1^r(N \nabla v)\cdot\nabla\zeta^*\,
dy,
\\
\mathbf
I_3=-\int_{Q^-_0}\partial_1^r(v_{r-1}N\nabla\varphi)\cdot\nabla
\zeta^*\,
dy,\\
\mathbf I_4=-\int_{Q^-_0}\partial_1^{r-1}(\nabla \varphi \cdot
N\nabla v_{r-1})\partial_1\zeta^*\, dy.
 \end{split}\end{equation}
Let us estimate step by step the quantities $\mathbf I_i$. We have
\begin{equation}\label{tran76aa}
    |\mathbf I_1|\leq c \|\partial_1^{r-1}(v_{r-1}
    N\nabla\varphi)\|_{W^{1/2,2}(\mathbb R)}\,\|\zeta\|_{W^{1/2,2}(\mathbb
    R)}.
\end{equation}
Recall that the function $\partial_1^{r-1}(v_{r-1}
    N\nabla\varphi)$ is compactly supported in $Q$. From this and
    the Poincare inequality we obtain
\begin{equation*}\begin{split}
\|\partial_1^{r-1}(v_{r-1}
    N\nabla\varphi)\|_{W^{1/2,2}(\mathbb R)}\leq c\|\partial_1^{r-1}(v_{r-1}
    N\nabla\varphi)\|_{W^{1,2}(Q_0)}\leq\\
c\|\nabla\partial_1^{r-1}(v_{r-1}
    N\nabla\varphi)\|_{L^2(Q_0)}.
\end{split}\end{equation*}
Noting that $N$ is independent of $y_2$, we get
\begin{equation}\label{tran76ab}\begin{split}
\|\partial_1^{r-1}(v_{r-1}
    N\nabla\varphi)\|_{W^{1/2,2}(\mathbb R)} \leq\\
c\|\partial_1^{r}(v_{r-1}
    N\nabla\varphi)\|_{L^2(Q_0)}+ c\|\partial_1^{r-1}(\partial_2v_{r-1}
    N\nabla\varphi)\|_{L^2(Q_0)}+\boldsymbol\mu,
\end{split}\end{equation}
where
\begin{equation}\label{tran76bb}
    \boldsymbol\mu=\|\partial_1^{r-1}(v_{r-1}
    N\partial_2\nabla\varphi)\|_{L^2(Q_0)}.
\end{equation}
Note  that the quantities $\partial_1^{r}(v_{r-1}
    N\nabla\varphi)$ and $\partial_1^{r-1}(\partial_2v_{r-1}
    N\nabla\varphi)$ are compactly supported in $Q=Q_r$. Since $a=1$ in $Q_0^-$, it follows
    from formulae \eqref{tran10} for $\mathcal B_{r-1}$,
    $\mathcal C_{r-1}$ and estimates \eqref{tran41}, \eqref{tran58}
    for $\mathcal B_{r-1}$,
    $\mathcal C_{r-1}$ in Lemma \ref{tran28} that
\begin{equation}\label{tran76bc}\begin{split}
 \|\partial_1^{r}(v_{r-1}
    N\nabla\varphi)\|_{L^2(Q_0)}+ \|\partial_1^{r-1}(\partial_2v_{r-1}
    N\nabla\varphi)\|_{L^2(Q_0)}\leq\\
    c\|\partial_1^{r}\mathcal B_{r-1}\|_{L^2(Q_0)}
    +\|\partial_1^{r-1}\mathcal C_{r-1}\|_{L^2(Q_0)}\leq \\
    c\big(1+\|\partial_1^r N\|_{L^2(-2\kappa, 2\kappa)}\,\big).
\end{split}\end{equation}
Let us estimate $\boldsymbol\mu$. We have
\begin{equation*}
\partial_1^{r-1}(v_{r-1}
    N\partial_2\nabla\varphi)=\sum c_{j,\alpha,\beta}\,\,\partial_1^j
    N\,\, \partial_1^\alpha v_{r-1}\,\,\partial_1^\beta
    \partial_2\nabla\varphi,
\end{equation*}
where the sum is taken over the set of indexes
\begin{equation*}
    j,\alpha, \beta\geq  0, \quad j+\alpha+\beta=r-1.
\end{equation*}
Since $N$ is uniformly bounded, we have
\begin{equation*}
\|\partial_1^j N\|_{L^\infty(-2\kappa,2\kappa)}\leq
c\big(1+\|\partial_1^{j+1}N\|_{L^2(-2\kappa, 2\kappa)}\,\big).
\end{equation*}
It follows  that
\begin{equation*}
    \boldsymbol\mu\leq\sum_{j=0}^{j=r-1}\big(1+\|\partial_1^{j+1}N\|_{L^2(-\kappa,
    \kappa)}\,\big)
    \,\,\sum_{\alpha+\beta=r-1-j}\|\partial_1^\alpha v_{r-1}\,\,\partial_1^\beta
    \partial_2\nabla\varphi\|_{L^2(Q_0)}.
\end{equation*}
Note that all derivatives of $\varphi$ are bounded. Thus we get
\begin{equation*}
\sum_{\alpha+\beta=r-1-j}\|\partial_1^\alpha
v_{r-1}\,\,\partial_1^\beta
    \partial_2\nabla\varphi\|_{L^2(Q_0)}\leq
    c\big(1+\|\partial_1^{r-1-j}v_{r-1}\|_{L^2(Q_0)}\,\big).
\end{equation*}
From this and estimate \eqref{tran60} in Proposition \ref{tran59} we
conclude that
$$
\sum_{\alpha+\beta=r-1-j}\|\partial_1^\alpha
v_{r-1}\,\,\partial_1^\beta
    \partial_2\nabla\varphi\|_{L^2(Q)}\leq
    c\big(1+\|\partial_1^{r-1-j}N\|_{L^2(-2\kappa, 2\kappa)}\,\big),
$$
which yields
\begin{equation*}
    \boldsymbol\mu\leq \sum_{j=0}^{r-1}\big(1+\|\partial_1^{j+1}N\|_{L^2(-2\kappa,
    2\kappa)}\,)\,\,
(1+\|\partial_1^{r-1-j}N\|_{L^2(-2\kappa, 2\kappa)}\,\big).
\end{equation*}
Since $N$ is uniformly bounded, we may apply the interpolation
inequality to obtain
\begin{equation*}\begin{split}
    \boldsymbol\mu\leq \sum_{j=0}^{r-1}\big(\,1+\|\partial_1^{r}N\|_{L^2(-2\kappa,
    2\kappa)}^{\frac{j+1}{r}}\,\big)\,\,
\big(\,1+\|\partial_1^{r}N\|_{L^2(-2\kappa,
2\kappa)}^{\frac{r-1-j}{r}}\,\big)\leq \\
c\big(1+\|\partial_1^{r}N\|_{L^2(-2\kappa,
    2\kappa)}\,\big).
\end{split}\end{equation*}
Combining this estimate with \eqref{tran76bc} and \eqref{tran76ab}
we obtain
\begin{equation*}\begin{split}
\|\partial_1^{r-1}(v_{r-1}
    N\nabla\varphi)\|_{W^{1/2,2}(\mathbb R)}\leq c\big(1+\|\partial_1^{r}N\|_{L^2(-2\kappa,
    2\kappa)}\,\big).
\end{split}\end{equation*}
From this and  expression \eqref{tran76} for $\mathbf I_1$ we
conclude that
\begin{equation}\label{tran77a}\begin{split}
|\mathbf I_1|\leq  c\,\big(1+\|\partial_1^{r}N\|_{L^2(-2\kappa,
    2\kappa)}\,\big)\,\,\|\zeta\|_{W^{1/2,2}(\mathbb R)}.
\end{split}\end{equation}

Let us estimate the integral $\mathbf I_2$. Note that
\begin{equation}\label{tran80}
    \partial_1^r( N\nabla v)= N\nabla\partial_1^r v
    +\mathcal A_r\text{~~in~~} Q^-_0,
\end{equation}
where $\mathcal A_r$ is given by equality \eqref{tran13}. It follows
from estimate \eqref{tran9} in Lemma \ref{tran28} and estimate
\eqref{tran60} in Proposition \ref{tran59} that
\begin{equation}\label{tran81}\begin{split}
\|\mathcal A_r\|_{L^2(Q)}\leq  c\big(\,1+\|\partial_1^r
N\|_{L^2(-\kappa,\kappa)}\,\big)+c\|\partial_1^r\nabla
v\|_{L^2(Q)}^\lambda\\ \leq c \big(\,1+\|\partial_1^r
N\|_{L^2(-\kappa,\kappa)}\big),
\end{split}\end{equation}
where $0\leq \lambda<1$. On the other hand, estimate \eqref{tran60}
in Proposition \ref{tran59} yields
$$
\|N\nabla\partial_1^r v\|_{L^2(Q)}\leq c\,\big(\,1+\|\partial_1^r
N\|_{L^2(-2\kappa,2\kappa)}\,\big).
$$
Combining  this estimate  with inequality \eqref{tran81} and
identity
 \eqref{tran80} we obtain
$$
 \|\partial_1^r( N\nabla v)\|_{L^2(Q^-)}\leq \|N\nabla\partial_1^r
 v\|_{L^2(Q^-)}
    +\|\mathcal A_r\|_{L^2(Q^-)}\leq c \big(\,1+\|\partial_1^r
N\|_{L^2(-\kappa,\kappa)}\,\big).
$$
Recall that the function $\zeta^*$ is extended by zero from $Q^-$ to
the strip $Q_\infty$. It follows from this and  the Cauchy
inequality that
\begin{equation*}\begin{split}
\Big|\,\int_{Q_0}\partial_1^r(N\nabla {v^-})\cdot\nabla\zeta^*\,
    dy\,\Big|\leq c \big(\,1+\|\partial_1^r
N\|_{L^2(-2\kappa,2\kappa)}\,\big)\,\|\nabla\zeta^*\|_{L^2(Q_\infty)}\\\leq
c \big(\,1+\|\partial_1^r
N\|_{L^2(-2\kappa,2\kappa)}\,\big)\,\|\zeta\|_{W^{1/2,2}(\mathbb
R)}.
 \end{split}\end{equation*}
This result along with expression \eqref{tran76a} for $\mathbf I_2$
implies the desired estimate
\begin{equation}\label{tran82}\begin{split}
|\mathbf I_2|\leq c \,\big(\,1+\|\partial_1^r
N\|_{L^2(-2\kappa,2\kappa)}\,\big)\,\,\|\zeta\|_{W^{1/2,2}(\mathbb
R)}.
 \end{split}\end{equation}
In order to estimate the integrals $\mathbf I_3$ and $\mathbf I_4$,
 note that
\begin{equation*}
\partial_1^r(v_{r-1}N\nabla\varphi)=\partial_1^r\mathcal B_{r-1},
\quad \partial_1^{r-1}(\nabla \varphi \cdot N\nabla
v_{r-1})=\partial_1^{r-1}\mathcal C_{r-1}\text{~~in~~} Q_0^-.
\end{equation*}
From this, estimates \eqref{tran41}-\eqref{tran58} and expression
\eqref{tran76a} for $\mathbf I_i$ we obtain
\begin{equation*}\begin{split}
|\mathbf I_3|+|\mathbf I_4|\leq\\ c(\|\partial_1^r\mathcal B_{r-1}
\|_{L^2(Q)}+\|\partial_1^{r-1}\mathcal C_{r-1} \|_{L^2(Q)})\,
\|\nabla\zeta^*\|_{L^2(Q)}\, \leq\\
c(1+\|\partial_1^r N\|_{L^2(-2\kappa,
2\kappa)})\,\|\nabla\zeta^*\|_{L^2(Q)}\leq\\
c\,\big(1+\|\partial_1^r N\|_{L^2(-2\kappa,
2\kappa)}\,\big)\,\,\|\zeta\|_{W^{1/2,2}(\mathbb R)}.
 \end{split}\end{equation*}
Combining this estimate with estimates \eqref{tran77a} and
\eqref{tran82} we obtain the estimate
$$
|\mathbf I_i|\leq  c\,\big(1+\|\partial_1^r N\|_{L^2(-2\kappa,
2\kappa)}\,\big)\,\,\|\zeta\|_{W^{1/2,2}(\mathbb R)}, \quad
i=1,\dots 4.
$$
Finally notice  that this estimate along with equality
\eqref{tran76} obviously yields desired estimate \eqref{tran74}.

In order to complete the proof of Proposition \ref{tran72}, it
remains  to obtain the  estimate,  similar to \eqref{tran74}, for
the function $\varphi\partial_N u^-$. Recall that $v=\varphi u$,
where $\varphi$ is an arbitrary function of the class
$C^\infty_0(Q)$. Now we specify $\varphi$. To this end, choose an
arbitrary function $\phi\in C^\infty_0(-\kappa, \kappa)$ and fix the
function $\iota(x_2)$, $x_2\in \mathbb R$, such that
$$
\iota\in C^\infty_0(\mathbb R), \quad \iota(x_2)=1\text{~~for~~}
|x_2|\leq \rho/3, \quad \iota(x_2)=0\text{~~for~~} |x_2|\geq
2\rho/3.
$$
Now set
$$\varphi\equiv\varphi_r=\phi(x_1)\iota(x_2).
$$
With this notation we have
\begin{equation*}
    \partial_N v^-=\phi\partial_N u^-+\partial_1 \phi
    N_{21}u^-=\phi\partial_N u^-+N_{21}\partial_1\phi\,
    v_{r-1}^-\text{~~on ~~}\ell,
\end{equation*}
since $u$ coincides with $v_{r-1}$ on the support of
$\nabla\varphi$. Note that
$$
N_{21}\partial_1\varphi\,
    v_{r-1}^-=v_{r-1}N\nabla\varphi\,\cdot \mathbf e_2\text{~~on~~} \ell.
$$
Thus we get
\begin{equation*}
 \varphi\partial_N u^-=   \partial_N v^-
-(v_{r-1}\partial_N\varphi)\text{~~on ~~}\ell.
\end{equation*}
In order to prove desired estimate \eqref{tran73} in Proposition
\ref{tran73} for $\partial_N u^-$, it suffices to show that the
inequality
\begin{equation}\label{tran201}
    \big|\, \int_{\mathbb R}\partial_1^r\,(\varphi\partial_N
    u^-)\,\zeta\, dy_1\big |\leq c\,(1+\|\partial_1^r
    N\|_{L^2(-2\kappa, 2\kappa)}\, )\, \|\zeta\|_{W^{1/2,2}(\mathbb
    R)}
\end{equation}
holds for all compactly supported functions $\zeta\in
W^{1/2,2}(\mathbb R)$. In its turn, estimate \eqref{tran201} is a
straightforward consequence of estimates
\begin{equation}\label{tran202}
    \big|\, \int_{\mathbb R}\partial_1^r\,(\partial_N
    v^-)\,\zeta\, dy_1\big |\leq c\,(1+\|\partial_1^r
    N\|_{L^2(-2\kappa, 2\kappa)}\, )\, \|\zeta\|_{W^{1/2,2}(\mathbb
    R)},
\end{equation}
\begin{equation}\label{tran203}
    \big|\, \int_{\mathbb R}\partial_1^r\,(v_{r-1}\partial_N
    \varphi)\,\zeta\, dy_1\big |\leq c\,(1+\|\partial_1^r
    N\|_{L^2(-2\kappa, 2\kappa)}\, )\, \|\zeta\|_{W^{1/2,2}(\mathbb
    R)}.
\end{equation}
It is easy to see that estimate \eqref{tran202} coincides with
estimate \eqref{tran74}. On the other hand, the left hand side of
\eqref{tran203} obviously equals the quantity $|\mathbf I_1|$
defined by \eqref{tran76a}. Therefore, estimate \eqref{tran203}
obviously follows from estimate \eqref{tran76aa} for $\mathbf I_1$.
It remains to note that \eqref{tran202} and \eqref{tran203} imply
desired estimate \eqref{tran201}. This completes the proof of
Proposition \ref{tran72}.









\section{Kohn-Vogelius functional. Proofs
 of Theorems \ref{normder1} and
\ref{bokr60}}\label{gradiend}
\renewcommand{\theequation}{5.\arabic{equation}}
\setcounter{equation}{0} Recall the formulation of the problem. Our
goal is to derive estimates of the gradient $dJ$ of the
Kohn-Vogelius functional $J$ in terms of the geometric
characteristics of the interface $\Gamma$. The results are based on
the normal derivatives estimate for solutions to the transmission
problem given in the previous section. These estimates establish the
dependence of the smoothness properties of solutions to a
transmission problem with respect to the smoothness properties of
the interface $\Gamma$. Recall the conditions
\textbf{H.1}-\textbf{H.3} imposed on $\Gamma$.

\begin{itemize}
\item[\textbf{H.1}] The Jordan curve $\Gamma\subset \Omega$ satisfies the
energy
%condition
inequality
$$
\frac{1}{2}\int_\Gamma |k^2|ds +\mathcal L\leq E_0.
$$

\item[\textbf{H.2}]
There is $\nu>0$ with the property
\begin{equation*}
    \text{\rm dist~}(\Gamma\setminus \Gamma_{3\kappa}, \,
    \Gamma_{2\kappa})\,\geq \nu
\end{equation*}
for every arc$\,\Gamma_{3\kappa}$ with  $\kappa$  defined by Lemma
\ref{bokr3a}.



\item[\textbf{H.3}] There is $\varrho>0$ such that
$$\text{dist~}(\Gamma,
\partial \Omega)>\varrho.
$$
\end{itemize}
By virtue of Corollary \ref{bokr3st}, every curve $\Gamma$
satisfying   conditions {\bf H.1}-{\bf H.3} is a Jordan curve of the
class $C^{1+\alpha}$, $0<\alpha<1/2$.~ $\Gamma$ splits the domain
$\Omega$ into two parts. The first $\Omega_i\Subset\Omega$~
(inclusion) is a simply connected domain with boundary $\Gamma$. The
second is the curvilinear annulus $\Omega_e=\Omega\setminus
\overline{\Omega_i}$ bounded by $\Gamma$ and $\partial\Omega$. For
simplicity, we will assume that $\partial\Omega$ is a Jordan curve
of the class $C^\infty$.

Next, recall that the piecewise constant function $a:\Omega\to
\mathbb R^+$ (conductivity) is defined by the equalities
\begin{equation*}
a=1\text{~~in~~}\Omega_e, \quad a=a_0\text{~~in~~}\Omega_i.
\end{equation*}







\subsection{Proof of Theorem \ref{normder1}}
 Let $w\in W^{1,2}(\Omega)$ be a weak solution to the equation
\begin{equation}\label{esty2}
    \text{~div~}(a\,\nabla w)=0\text{~~in~~} \Omega.
\end{equation}
Denote by $w^-$ and $w^+$ the restrictions of $w$ onto subdomains
$\Omega_e$ and $\Omega_i$,
\begin{equation}\label{esty3}
w^-:=w\text{~~in~~}\Omega_e, \quad w^+:=w\text{~~in~~}\Omega_i.
\end{equation}
If $\Gamma$ is sufficiently smooth, then $w$ is continuous on
$\Gamma$. In other words, $w^-=w^+$ on $\Gamma$. However, the normal
derivative of $w$ has a jump across $\Gamma$. The following remark
is important for the further analysis. Set
\begin{equation}\label{esty500}
\partial_n w^-=\nabla w^-\cdot n, \quad \partial_n w^+=\nabla
w^+\cdot n\text{~~on~~} \Gamma.
\end{equation}
Our task is to to prove the estimate \eqref{esty6}:
\begin{equation*}
    \|\partial_n w^\pm\|_{H^{m+1/2}(\Gamma)}\leq c\,(\,1+\|
    \partial_s^m
    k\|_{L^2(\Gamma)}\,)\,\|w\|_{W^{1,2}(\Omega)},
    \end{equation*}
where $m\geq 0$ is an arbitrary integer, $c$ depends only on $m$ and
on the constants $E_0$, $\nu$, $\rho$ in Conditions
\textbf{H.1}-\textbf{H.3}. We split the proof of estimate
\eqref{esty6} into three steps. First, we define a standard
neighborhood of an arbitrary point $z\in \Gamma$ and the special
mapping which takes diffeomorphically the standard neighborhood onto
 a rectangle. Next we employ Proposition \ref{tran72} in Section
 \ref{model} in order to obtain the local version of estimate
 \eqref{esty6}. Finally, we use the local estimate and a partition of
 unity to complete the proof of \eqref{esty6}.



\subsubsection{Standard neighborhood and standard mapping}
 Note that the immersion $f$ and  the curve $\Gamma=f(\mathbb
S^1)$ satisfy all conditions of Lemma \ref{bokr3a}. Choose an
arbitrary $z\in \Gamma$  and consider the subarc $\Gamma_{2\kappa}$
of the arc $\Gamma_{3\kappa}$ defined by Lemma \ref{bokr3a}. It
follows from this lemma that $\Gamma_{2\kappa}$ admits the
representation
\begin{equation*}
    \Gamma_{2\kappa}:\, x_2=\eta(x_1), \quad x_1\in  (-\gamma,
    \delta)
\end{equation*}
with positive $\gamma$, $\delta$, depending only on $\kappa$. Next,
assume that $\Gamma$ has no self intersections and is compactly
embedded into a bounded domain $\Omega\subset \mathbb R^2$.  Set
\begin{equation}\label{immer3}
    2\rho=\min\big\{\nu, \text{~dist~}(\Gamma,\partial\Omega)\big\}>0,
\end{equation}
where $\nu>0$ is given by Condition \textbf{H.2}.

\begin{definition}\label{immer3b} Under the above assumptions, the
standard neighborhood $\Sigma_{2\kappa}$  of the point $z$ is the
curvilinear quadrangle defined in the Cartesian coordinates system
associated with $z$ by the equalities
\begin{equation}\label{immer3a}
    \Sigma_{2\kappa}=\big\{ x=(x_1,x_2):\, -\gamma<x_1<\delta, \quad
    -2\rho+\eta(x_1)<x_2<2\rho+\eta(x_1)\big\}.
\end{equation}
\end{definition}



Our next task is to define the special mapping, which takes the
standard neighborhood onto the rectangle. To this end introduce new
variables
\begin{equation}\label{immer4}
y_1=s(x_1), \quad y_2= x_2-\eta(x_1).
\end{equation}
It is easy to see that the mapping $y=y(x)$ takes diffeomorphically
the standard neighborhood $\Sigma_{2\kappa}$ onto the rectangle
\begin{equation}\label{immer5}
Q_0=(-2\kappa,2\kappa)\times(-2\rho,2\rho).
\end{equation}
Introduce the matrices $M$ and  $N$ defined by the equalities
\begin{equation}\label{immer7}
M:=y'(x)=\left(
\begin{array}{cc}
\displaystyle {\sqrt{1+{\eta'}^2}}&\displaystyle { 0}
\\ \displaystyle {- \eta'} &  \displaystyle {1
 }
  \end{array}
\right), \quad N=(\det M)^{-1}\, M\, M^\top.
\end{equation}
Notice that these matrices depend only on $x_1$ and hence only on
$s\in(-2\kappa,2\kappa)$. Introduce the function $\Theta:(-2\kappa,
2\kappa)\to \mathbb R$ defined by the equalities
\begin{equation}\label{immer8}
\Theta(s)=\arctan \eta'(x_1(s)), \quad \Theta(0)=0,
\end{equation}
in the Cartesian coordinates system associated with the origin $z$.
Recall that $x=0$ and $s=0$ at the chosen point $z$.

\begin{lemma}\label{immer9}
Under the above assumptions, we have
\begin{equation}\label{immer10}
    \Theta\in W^{1,2}(-2\kappa, 2\kappa), \quad |\Theta|\leq
    \frac{\pi}{18},
\end{equation}
\begin{equation}\label{immer11}
    \tau(s)=\big(\cos\Theta, \,\,\sin\Theta), \quad n(s)=(-\sin\Theta,
    \,\,\cos\Theta), \quad k(s)=\Theta'(s)\, n(s).
\end{equation}
Moreover, the matrices $M$ and $N$ admit the representation
\begin{equation}\label{immer12}
M=\left(
\begin{array}{cc}
\displaystyle {\frac{1}{\cos\Theta}}&\displaystyle { 0}
\\ \displaystyle {- \tan \Theta} &  \displaystyle {1
 }
  \end{array}
\right), \quad N=\frac{1}{\cos\Theta}\left(
\begin{array}{cc}
\displaystyle {1}&\displaystyle {- \sin\Theta}
\\ \displaystyle {- \sin\Theta} &  \displaystyle {1
 }
  \end{array}
\right).
\end{equation}
\end{lemma}
\begin{proof}Introduce the function
$$
\lambda(s(x_1))=\eta(x_1), \quad s\in (-2\kappa, 2\kappa).
$$
We have
$$
\lambda'(s)\sqrt{1+{\eta'(x_1)}^2}\, =\eta'(x_1)\text{~~and hence~~}
\lambda'(s)^2=\eta'(x_1)^2-\lambda'(s)^2\,{\eta'(x_1)}^2,
$$
which gives
\begin{equation}\label{immer13}
\eta'=\frac{\lambda'}{\sqrt{1-{\lambda'}^2}}, \quad
1+{\eta'}^2=\frac{1}{1-{\lambda'}^2}.
\end{equation}
In particular, we have
\begin{equation}\label{immer14}
M=\left(
\begin{array}{cc}
\displaystyle {\frac{1}{\sqrt{1-{\lambda'}^2}}}&\displaystyle { 0}
\\ \displaystyle {- \frac{\lambda'}{\sqrt{1-{\lambda'}^2}}} &  \displaystyle {1
 }
  \end{array}
\right).
\end{equation}
Next notice that
\begin{equation}\label{immer15}
\tau=\frac{1}{\sqrt{1+{\eta'}^2}}\, (1,
\eta')=(\sqrt{1-{\lambda'}^2},\, \lambda')=(\cos\Theta, \,
\sin\Theta)
\end{equation}
with
$$
\Theta=\arctan \eta'=\arcsin \lambda'.
$$
We thus get the following formulae for the curvature vector $k$ and
the normal $n$
$$
k=\partial_s\tau =\Theta'(s) n, \quad n=(-\sin\Theta, \cos\Theta).
$$
 Identity \eqref{immer15} implies that the
 matrices
$M$ and $N$ admit representation \eqref{immer13}. It remains to note
that $|\tan\Theta|=|\eta'|\leq 1/6$, which yields the estimate
$|\Theta|\leq\pi/18$.
\end{proof}

\begin{corollary}\label{immer16}
Under the assumptions of Lemma \ref{immer9},  the estimate
\begin{equation}\label{immer17}
    \|\partial_s^m N\|_{L^2(-2\kappa, 2\kappa)}\leq
    c(m)(1+\|\partial_s^{m-1}k\|_{L^2(-2\kappa, 2\kappa)})
\end{equation}
holds for every integer $m\geq 1$.
\end{corollary}
\begin{proof}
Since $|\Theta|\leq \pi/18$, it follows from the estimates of
composite functions in Sobolev spaces that
$$
\|\partial_s^m N\|_{L^2(-2\kappa, 2\kappa)}\leq
    c(C_N)\big(1+\|\partial_s^{m}\Theta\|_{L^2(-2\kappa, 2\kappa)}\,\big ),
$$
where
$$
C_N=\sum_{k=0}^m\sup\limits_{|\Theta|\leq \pi/18} |\partial_\Theta^k
N(\Theta)|.
$$
It remains to note that $\partial_s^m\Theta=\partial_s^{m-1}k.$
\end{proof}




\subsection{Local estimates} In this paragraph we prove the local
estimates of the normal derivative of a weak solution to equation
\eqref{esty2}. The result is given by the following proposition. Fix
an arbitrary point $z\in \Gamma$. Without loss of generality we may
assume that the arc-length variable $s$ equals zero at $z$. Let
$\kappa$, depending only on the constant $E_0$ in Condition
\textbf{H.1}, be given by Lemma \ref{bokr3a}. Choose an arbitrary
function $\phi \in C^\infty_0(-\kappa, \kappa)$. Furthermore, assume
that the functions $\phi(s)$ and $\phi(s)\partial_n w^{\pm}(s)$ are
extended by zero to the real axis $\mathbb R$.



\begin{proposition}\label{esty5}
Under the above assumptions, the estimate
\begin{equation}\label{esty6a}
    \|\phi\,\partial_n w^\pm\|_{W^{m+1/2,2}(\mathbb R)}\leq c\big(\,1+\|
    \partial_s^m
    k\|_{L^2(-2\kappa,2\kappa)}\,\big)\,\|w\|_{W^{1,2}(\Omega)}
\end{equation}
 holds for every integer $m\geq 0$. Here $c$ depends only on $m$ and
on the constants $E_0$,$\nu$, $\rho$ in Conditions {\rm
\textbf{H.1}-\textbf{H.3}} of Theorem \ref{bokr60}.
\end{proposition}

\begin{proof}
Notice that estimate \eqref{esty6a} is invariant with respect to
dilatation $w$. Hence without loss of generality we may assume that
$\|w\|_{W^{1,2}(\Omega)}=1$. For an arbitrary fixed $z\in \Gamma$,
denote by $\Sigma_{2\kappa}$ the standard neighborhood determined by
Definition \ref{immer3b}. Split $\Sigma_{2\kappa}$ into two disjoint
parts $\Sigma_{2\kappa}^-$ and $\Sigma_{2\kappa}^+$ defined by the
equalities
\begin{equation}\label{esty8}\begin{split}
    \Sigma_{2\kappa}^-=\big\{ x=(x_1,x_2):\, -\gamma<x_1<\delta, \quad
\eta(x_1)-2\rho<x_2<\eta(x_1)\,\big\}, \\
    \Sigma_{2\kappa}^+=\big\{ x=(x_1,x_2):\, -\gamma<x_1<\delta, \quad
    \eta(x_1)<x_2<2\rho+\eta(x_1)\,\big\}.
\end{split}\end{equation}
Here $(x_1,x_2)$ is the local Cartesian coordinates system with the
origin at $z$, defined in Lemma \ref{bokr3a}. Notice that the
ordinate axis $x_2$ is directed inside $\Omega_i$, which yields
$$
\Sigma_{2\kappa}^-=\Sigma_{2\kappa}\cap \Omega_e, \quad
\Sigma_{2\kappa}^+=\Sigma_{2\kappa}\cap \Omega_i, \quad
\Sigma_{2\kappa}=\Sigma_{2\kappa}^-\cup\Sigma_{2\kappa}^+\cup\Gamma_{2\kappa}.
$$
In particular, the coefficient $a$ equals $1$ in
$\Sigma_{2\kappa}^-$ and equals $a_0$ in $\Sigma_{2\kappa}^+$.
 The function $w$ serves as a solution to equation
\eqref{esty2} and  the integral identity
\begin{equation}\label{esty9}
\int_{\Sigma_{2\kappa}}a\nabla w\cdot\nabla\zeta\, dx=0
\end{equation}
holds for all  $\zeta\in W^{1,2}_0(\Sigma_{2\kappa})$. The standard
change of variables
\begin{equation*}
y_1=s(x_1), \quad y_2= x_2-\eta(x_1).
\end{equation*}
takes diffeomorphically  the standard neighborhood
$\Sigma_{2\kappa}$ onto the rectangle
$$Q_0=(-2\kappa,2\kappa)\times (-2\rho,2\rho).$$
In its turn, the standard change of the variables \eqref{immer4}
takes diffeomorphically curvilinear quadrangles
$\Sigma_{2\kappa}^\pm$ onto the rectangles
$$
Q_0^-= (-2\kappa,2\kappa)\times (-2\rho,0), \quad
Q_{2\kappa}^+=(-2\kappa,2\kappa)\times (0,2\rho).
$$
We have
$$
Q_0=Q_0^- \cup Q_0^+\cup \ell, \text{~where ~~}\ell= (-2\kappa,
2\kappa)\times \{0\}.
$$
Now set
\begin{equation}\label{esty10}
u(y)=w(x(y)), \quad u^{\pm}(y)=w^\pm(x(y))= u(y)\Big|_{Q_0^\pm}.
\end{equation}
Recall that $w$ serves as a weak solution to equation \eqref{esty2}
in the standard neighborhood $\Sigma_{2\kappa}$. In particular,  it
satisfies integral identity \eqref{esty9}. Notice that
$$
\nabla_x w(x(y)) = M^\top \nabla_y u(y), \quad dx=(\det\, M)^{-1}
dy.
$$
The change of  variables $x\to y$ in  \eqref{esty9} leads to the
following integral identity for the function $u:Q_0\to \mathbb R$,
\begin{equation}\label{esty11}
\int_{\Sigma_{2\kappa}}a\, N\nabla u\cdot\nabla\zeta\,
dy=0\text{~~for all~~}\zeta\in W^{1,2}_0(Q_0).
\end{equation}
 Here the matrix $N$ is given by Lemma \ref{immer9}. The conormal
derivative $\partial_N u^-$ on the segment $\ell$ is defined by the
equality
\begin{equation}\label{esty13}
    \partial_N u^-=N_{21}\partial_{y_1} u^-+N_{22}\partial_{y_2}u^-=
    \frac{1}{\cos \Theta}\,\big(-\partial_{y_1}u^-\,\sin\Theta
    +\partial_{y_2}u^-\big),
\end{equation}
where $\Theta(s)$ is given by Lemma \ref{immer9}. It is easily seen
that $N$ and $u$ meet all requirements of Proposition \ref{tran72}.
Now choose an arbitrary $\phi\in C^\infty_0(-\kappa,\kappa)$ and
assume that the function $\phi\partial_N u^-(s)$ extended by $0$ to
the real axis. Applying estimate \eqref{tran73} in Proposition
\ref{tran72} and estimate \eqref{immer17} in Corollary \ref{immer16}
we conclude that  the inequality
\begin{equation}\label{esty15}
    \|\phi\partial_N u^-\|_{W^{r-1/2}(\mathbb R)}\leq c\,
    \big(1+\partial_s^r N\|_{L^2(-2\kappa,2\kappa)}\,\big)\leq c\,
    \big(1+\|\partial_s^{r-1}k\|_{L^2(-2\kappa, 2\kappa)}\,\big)
\end{equation}
holds for every integer $r\geq 1$.

Recall that $s\equiv y_1$. Next, formulae \eqref{immer11} and
\eqref{immer12} in Lemma \ref{immer9} along with formula
\eqref{esty13} imply
\begin{equation}\label{esty14}\begin{split}
n(s)\cdot\nabla_x w^-(x(y))\equiv n(s)M^\top\cdot \nabla
v^-=\\
(-\sin\Theta, \cos\Theta)\cdot \left(
\begin{array}{cc}
\displaystyle {\frac{1}{\cos\Theta}}&\displaystyle {- \tan \Theta}\\
\displaystyle {0}&\displaystyle {1}
 \end{array}
\right)\nabla u=\\
\frac{1}{cos\Theta}(-\sin\Theta\,\partial_{y_1}u+\partial_{y_2} u)=
\partial_N u^-\text{~~on~~}\ell.
\end{split}\end{equation} It
follows that for every $\phi\in C^\infty_0(-\kappa,\kappa)$, we
 have
$$
\phi(s)\partial_n w^-(s)=\phi(s)\partial_N u^-(s)\text{~~for~~} s\in
(-2\kappa,2\kappa).
$$
Hence  desired estimate \eqref{esty6a} is the straightforward
consequence of \eqref{esty15}. This completes the proof of
Proposition \ref{esty5}.
\end{proof}








\subsection{Globalization} Now we employ Proposition \ref{esty5} in
order to complete the proof of Theorem \ref{normder1}. To this end,
we use the method of partition  of unity.


{\bf The partition of unity.} Let $\Gamma$ satisfies all conditions
of Theorem \ref{normder1} and $\kappa\in (0,\mathcal L/2)$ be given
by Lemma \ref{bokr3a}. Recall that $\mathcal L$ and $\kappa$ depends
only on the constant $E_0$ in the energy constraint \textbf{H.1}.
Choose a finite collection of points $z_k\in \Gamma$ with the
arc-length coordinates $s_k$ such that
\begin{equation}\label{unit1}
s_k=\frac{k}{N}\,\mathcal L,\quad 0\leq k\leq N-1,
\end{equation}
where $N$ is an arbitrary integer satisfying the condition
$$
\frac{1}{N} <\frac{\kappa}{4}.
$$
Now choose an arbitrary function $\psi\in C^{\infty}_0(-\mathcal
L/2, \mathcal L/2)$ with the properties
\begin{equation}\label{unit2}
\psi(s)\geq 0, \quad \psi(s)=0\text{~~for~~} |s|\geq \kappa, \quad
\psi(s)=1\text{~~for~~} |s|\leq \kappa/2.
\end{equation}
We will  consider $\psi(s)$ as a function defined on $\Gamma$. In
other words, it can be regarded as $\mathcal L$-periodic function
defined on $\mathbb R$. For every integer $k\in [0,N-1]$, define the
function
$$\psi_k(s)=\psi(s-s_k).$$
The function $\psi_k:\Gamma\to \mathbb R$ is compactly supported on
the arc of the length $2\kappa$ centered at $z_k$. Moreover,
$\psi_k=1$ on the arc of length $\kappa$ centered at  $z_k$.  These
arcs cover the  curve $\Gamma$. It is easily seen that for every
$z\in \Gamma$, at least one of the functions $\psi_k$ equals $1$ in
a neighborhood of $z$.

Now set
\begin{equation}\label{unit3}
    \phi_k=\frac{\psi_k}{\sum\limits_{j=0}^{N-1} \psi_j}.
\end{equation}
It is clear that every nonnegative function $\phi_k\in
C^\infty(\Gamma)$ is compactly supported in the arc of length
$2\kappa$ centered at $z_k$ and
\begin{equation}\label{unit4}
    \sum\limits_{k=0}^{N-1} \phi_k=1.
\end{equation}
Introduce the functions $\omega_k$ with the properties
\begin{equation}\label{unit5}
    \omega_k=\phi_k \partial_n w^-, \quad \partial_n w^-=\sum\limits_{k=0}^{N-1} \omega_k.
\end{equation}
It is clear that for every $m\geq 0$, we have
\begin{equation}\label{unit6}
    \|\partial_n w^-\|_{H^{m+1/2}(\Gamma)}\leq
    \sum\limits_{k=0}^{N-1} \|\omega_k\|_{H^{m+1/2}(\Gamma)}.
\end{equation}
See Subsection \ref{sobolevfun} for the definition of spaces
$H^r(\Gamma)=H^r_\sharp$.

{\bf Global estimates of the normal derivatives.} It follows from
\eqref{unit6} that it suffices to estimate $\omega_k$ in the space
$H^{m+1/2}(\Gamma)$. To this end we use the following construction.
Choose an arbitrary compactly supported function
$F:(-\kappa,\kappa)\to \mathbb R$.  There are two ways to extend $F$
to the real line. The first way is to extend $F$ by zero to $\mathbb
R$. We denote this extension by $\overline{F}$,
$$
\overline{F}(s)=F(s)\text{~~for~~} |s|<\kappa, \quad
\overline{F}=0\text{~~otherwise}.
$$
The second way is to extend $F$ $\mathcal L$-periodically to
$\mathbb R$. We denote this extension by $F_\sharp$. These extension
are connected by the relation
\begin{equation}\label{unit7}
    F_\sharp(s)=
    \sum\limits_{k=-\infty}^{\infty} \overline{ F} (s+k\,\mathcal L).
\end{equation}
The following lemma establishes estimates of  Sobolev norms of
$\overline{F}$ and $F_\sharp$.
\begin{lemma}\label{unit8}
There is the constant $c$ depending only on $\mathcal L$ and $m$
such that
\begin{equation}\label{unit9}
    \| F_\sharp\|_{H^{m+1/2}_\sharp}\leq c\|\overline
    F\|_{H^{m+1/2}(\mathbb R)}\text{~~ for all ~~} m\geq 0.
\end{equation}
The norms in the spaces $H^{m+1/2}_\sharp$ and $H^{m+1/2}(\mathbb
R)$ are defined in Subsection \ref{sobolevfun}.
\end{lemma}
\begin{proof} The proof is given in Subsection \ref{extension}.
\end{proof}

We are now in a position to complete the proof of Theorem
\ref{normder1}. To this end, it suffices to prove that every
function $\omega_k$ defined by \eqref{unit5} admits the estimate
\begin{equation}\label{unit10}
    \|\omega_k\|_{H^{m+1/2}(\Gamma)}\leq c\,
    \big(1+\|\partial_s^m\,k\|_{L^2(\Gamma)}\,\big)\,\,\|w\|_{W^{1,2}(\Omega)}.
\end{equation}
Fix an arbitrary integer $k\in [0, N-1]$ and the corresponding point
$z_k=f(s_k)$. After the shift of the coordinate $s$ we may assume
that $s_k=0$.  Let the arc $\Gamma_k$ centered at $z_k$ is defined
by Lemma \ref{bokr3a} with $z$ replaced by
 $z_k$. Introduce the
function $F:\Gamma_\kappa \to \mathbb R$ given by the equality
\begin{equation}\label{unit11}
    F=\phi_k(s)\,\partial_n w^- (s), \quad s\in (-\kappa, \kappa).
\end{equation}
Notice that $\phi_k\in C^\infty_0(-\kappa, \kappa)$. Let
$\overline{F}$ be the extension by $0$ to the real line.  It follows
from  estimate \eqref{esty6a} in Lemma \ref{unit8} that
\begin{equation}\label{unit12}
    \|\overline{F}\|_{H^{m+1/2}(\mathbb R)}\leq c\,\big(1+\|\partial_s^m
    k\|_{L^2(\Gamma)}\,\big)\,\, \|w\|_{W^{1,2}(\Omega)}.
\end{equation}
On the other hand, relations \eqref{unit5} and \eqref{unit7} imply
the equality
$$
F_\sharp= \omega_k\text{~~on~~}\Gamma.
$$
From this, estimate \eqref{unit12}, and estimate \eqref{unit9} in
Lemma \ref{unit8} we finally obtain
\begin{equation*}\begin{split}
    \|\omega_k\|_{H^{m+1/2}(\Gamma)}=
    \|F_\sharp\|_{H^{m+1/2}(\Gamma)}\leq
    c\|\overline{F}\|_{H^{m+1/2}(\mathbb R)}\leq\\
c\,\big(1+\|\partial_s^m
    k\|_{L^2(\Gamma)}\,\big)\,\, \|w\|_{W^{1,2}(\Omega)},
\end{split}\end{equation*}
which yields desired estimate \eqref{unit10}. This completes the
proof of Theorem \ref{normder1}.








\subsection{Proof of Theorem \ref{bokr60}}
Recall that the Kohn-Vogelius functional is defined by the formula
\begin{equation}\label{shopt2norm}
    J(\Gamma)=\int_\Omega a\nabla(v-w)\cdot\nabla(v-w)\,dx.
\end{equation}
Here   $v,w:\Omega\to \mathbb R$ satisfy the equations and boundary
conditions
\begin{align}\notag
\text{div~}a\nabla v  & = 0 & \text{div~}a\nabla w  & = 0 &
\text{in~~} & \Omega,
\\   \label{shopt111norm}
a\nabla v\cdot  n & = g & w & = h & \text{on~~} & \partial \Omega.
\end{align}
The gradient $dJ$ is defined by equality \eqref{gala11},
\begin{equation}\label{gala115}
d J= 2\big(a\nabla v\cdot n\big[\partial_n v\big]- a\nabla w\cdot
n\big[\partial_n w\big])n-\big[a\nabla v\cdot\nabla v-a\nabla
w\cdot\nabla w\big]n.
\end{equation}
Let a curve $\Gamma$ satisfies conditions \textbf{H.1}-\textbf{H.3}
and the curvature  $k\in H^r(\Gamma)$ with an integer $r\geq 0$.
 It is necessary to prove that for every $\beta\in [0,1/2)$, the gradient $dJ$
 admits the estimate
\begin{equation}\label{bokr61a}
    \| dJ\|_{H^{r+\beta}(\Gamma}\leq c(1+\|\partial_s^r
    k\|_{L^2(\Gamma)}),
\end{equation}
where the  constant $c$ depends on $r$, $\beta$, and constants
$E_0$, $\nu$, $\varrho$ in conditions \textbf{H.1}-\textbf{H.3}.
Note that estimate \eqref{bokr61a} with a suitable $\beta$ and the
embedding theorems imply the inequality
\begin{equation}\label{bokr61bb}
    \| \partial_s^r dJ\|_{L^q(\Gamma)}\leq c(1+\|\partial_s^r
    k\|_{L^2(\Gamma)}),
\end{equation}
which holds for every $q\in [1,\infty)$. In this case the constant
$c$ depends in addition on $q$. The rest of the section is devoted
to the proof of  estimate  \eqref{bokr61a}. The key observation  is
that this estimate is straightforward consequence of Theorem
\ref{normder1}. It follows from inequalities \eqref{grad5} that the
solutions  $v$ and $w$ to problems \eqref{shopt111norm} admit the
estimate
\begin{equation*}
    \|v\|_{W^{1,2}(\Omega)}+\|w\|_{W^{1,2}(\Omega)}\leq c(g,h).
\end{equation*}
Hence they meet all requirements of Theorem \ref{normder1}. Applying
this theorem we conclude that estimates
\begin{equation}\label{gala541}
    \|\partial_n v^\pm\|_{H^{m+1/2}(\Gamma)}+\|\partial_n
    w^\pm\|_{H^{m+1/2}(\Gamma)}\leq c\,\big(\,1+\|\partial^m
    k\|_{L^2(\Gamma)}\,\big)
\end{equation}
holds true for all integers $m\geq 0$. Next, it follows from
\eqref{gala115} that $dJ$ is a quadratic form of the normal
derivatives $\partial_n v^\pm$ and $\partial_n w^{\pm}$. Hence it
suffices to estimates the products $\partial_n v^\pm\,\partial_n
w^\pm$ and $\partial_n v^\pm\,\partial_n w^\mp$. Let us estimate
$\partial_n v^-\partial_n w^-$. The  proofs of estimates of other
terms are similar. It is necessary to show that
\begin{equation}\label{gala542}
    \|\partial_n v^-\partial_n w^-\|_{H^{m+\beta}(\Gamma)}\leq c\,\big(\,1+\|\partial^m
    k\|_{L^2(\Gamma)}\,\big)
\end{equation}
for all integers $m\geq 1$ and for all $\beta\in [0,1/2)$. It
follows from the Moser inequality that
\begin{equation}\label{gala543}\begin{split}
    \|\partial_n v^-\partial_n w^-\|_{H^{m+\beta}(\Gamma)}\leq c\,
 \|\partial_n v^-\|_{L^\infty(\Gamma)}\,\|\partial_n
 w^-\|_{H^{m+\beta}(\Gamma)}\\
+c\,\|\partial_n w^-\|_{L^\infty(\Gamma)}\,\|\partial_n
v^-\|_{H^{m+\beta}(\Gamma)}. \end{split}\end{equation} By virtue of
 the embedding theorem, the estimates
$$
 \|\partial_n v^-\|_{L^\infty(\Gamma)}\leq c(\sigma)\,\|\partial_n
 v^-\|_{H^{1/2+\sigma}(\Gamma)}, \quad
 \|\partial_n w^-\|_{L^\infty(\Gamma)}\leq c(\sigma)\,\|\partial_n
 w^-\|_{H^{1/2+\sigma}(\Gamma)}
$$
hold for every $\sigma>0$. Applying the interpolation inequality to
the right hand sides of these estimates we obtain
$$
\|\partial_n
 v^-\|_{H^{1/2+\sigma}(\Gamma)}\leq c\,\|\partial_n
 v^-\|_{H^{1/2}(\Gamma)}^{1-\frac{\sigma}{m}}\,\,\|\partial_n
 v^-\|_{H^{m+1/2}(\Gamma)}^{\frac{\sigma}{m}},
 $$
 $$ \|\partial_n
 w^-\|_{H^{1/2+\sigma}(\Gamma)}\leq c\,\|\partial_n
 w^-\|_{H^{1/2}(\Gamma)}^{1-\frac{\sigma}{m}}\,\,\|\partial_n
 w^-\|_{H^{m+1/2}(\Gamma)}^{\frac{\sigma}{m}}.
$$
Inequality  \eqref{gala541} with $m=0$ implies
$$
\|\partial_n
 v^-\|_{H^{1/2}(\Gamma)}+\|\partial_n
 w^-\|_{H^{1/2}(\Gamma)}\leq c.
 $$
Hence  for every $m\geq 1$, we have
\begin{equation}\label{gala544}\begin{split}
\|\partial_n
 v^-\|_{L^{\infty}(\Gamma)}\leq c\,\,\|\partial_n
 v^-\|_{H^{m+1/2}(\Gamma)}^{\frac{\sigma}{m}}, \\
  \|\partial_n
 w^-\|_{L^{\infty}(\Gamma)}\leq c\,\|\partial_n
 w^-\|_{H^{m+1/2}(\Gamma)}^{\frac{\sigma}{m}}.
\end{split}\end{equation} Next,
the interpolation inequality implies
$$
\|\partial_n
 v^-\|_{H^{m+\beta}(\Gamma)}\leq c\,\|\partial_n
 v^-\|_{H^{1/2}(\Gamma)}^{\frac{1}{m}(\frac{1}{2}-\beta)}\,\,\,\|\partial_n
 v^-\|_{H^{m+1/2}(\Gamma)}^{1-\frac{1}{m}(\frac{1}{2}-\beta)},
 $$
 $$
\|\partial_n
 w^-\|_{H^{m+\beta}(\Gamma)}\leq c\,\|\partial_n
 w^-\|_{H^{1/2}(\Gamma)}^{\frac{1}{m}(\frac{1}{2}-\beta)}\,\,\,\|\partial_n
 w^-\|_{H^{m+1/2}(\Gamma)}^{1-\frac{1}{m}(\frac{1}{2}-\beta)}.
$$
It follows that
\begin{equation*}\begin{split}
\|\partial_n
 v^-\|_{H^{m+\beta}(\Gamma)}\leq c\,\|\partial_n
 v^-\|_{H^{m+1/2}(\Gamma)}^{1-\frac{1}{m}(\frac{1}{2}-\beta)},\\
\|\partial_n
 w^-\|_{H^{m+\beta}(\Gamma)}\leq c\,\,\|\partial_n
 w^-\|_{H^{m+1/2}(\Gamma)}^{1-\frac{1}{m}(\frac{1}{2}-\beta)}.
\end{split}\end{equation*}
Substituting these inequalities along with inequalities
\eqref{gala544} into \eqref{gala543} we obtain
\begin{equation}\label{gala545}\begin{split}
    \|\partial_n v^-\partial_n w^-\|_{H^{m+\beta}(\Gamma)}\leq c\,
\,\|\partial_n
 v^-\|_{H^{m+1/2}(\Gamma)}^{\frac{\sigma}{m}}\,\,
\|\partial_n
 w^-\|_{H^{m+1/2}(\Gamma)}^{1-\frac{1}{m}(\frac{1}{2}-\beta)}\\
 +\|\partial_n
 w^-\|_{H^{m+1/2}(\Gamma)}^{\frac{\sigma}{m}}\,\,
\|\partial_n
 v^-\|_{H^{m+1/2}(\Gamma)}^{1-\frac{1}{m}(\frac{1}{2}-\beta)}.
\end{split}\end{equation}
Now set
$$
\sigma=\frac{1}{2}-\beta, \quad \lambda=\frac{\sigma}{m}.
$$
We have
$$
\frac{1}{m}\big(\frac{1}{2}-\beta\big)=\lambda, \quad
1-\frac{1}{m}(\frac{1}{2}-\beta)=1-\lambda.
$$
From this and \eqref{gala545} we conclude that
\begin{equation*}\begin{split}
    \|\partial_n v^-\partial_n w^-\|_{H^{m+\beta}(\Gamma)}\leq c\,
\,\|\partial_n
 v^-\|_{H^{m+1/2}(\Gamma)}^{\lambda}\,\,
\|\partial_n
 w^-\|_{H^{m+1/2}(\Gamma)}^{1-\lambda}\\
 +\|\partial_n
 w^-\|_{H^{m+1/2}(\Gamma)}^{\lambda}\,\,
\|\partial_n
 v^-\|_{H^{m+1/2}(\Gamma)}^{1-\lambda}.
\end{split}\end{equation*}
Applying the Young inequality we arrive at the estimate
\begin{equation*}\begin{split}
    \|\partial_n v^-\partial_n w^-\|_{H^{m+\beta}(\Gamma)}\leq c\,
\,\|\partial_n
 v^-\|_{H^{m+1/2}(\Gamma)}+c \,\,
\|\partial_n
 w^-\|_{H^{m+1/2}(\Gamma)}.
\end{split}\end{equation*}
It remains to note this inequality along with inequality
\eqref{gala541} obviously imply  desired estimate \eqref{gala542}.
This completes the proof of Theorem \ref{bokr60}.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%% SECTION 5 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\section{A priori estimates. Proof of Theorem \ref{theoremsobolev}
}\label{appendixb}
\renewcommand{\theequation}{6.\arabic{equation}}
\setcounter{equation}{0} In  this section we prove Theorem
\ref{theoremsobolev}. To this end, we have to derive the Sobolev a
priori estimates of  solutions to the Cauchy problem
\begin{equation}\label{appb1}\begin{split}
\partial_t f+\nabla_s^2\, k+\frac{|k|^2}{2}k-k+dJ=0\text{~~in~~}
(0,T)\times\mathbb S^1,\\
f(0,\theta)=f_0(\theta), \quad \theta\in\mathbb S^1
\end{split}\end{equation} for an  immersion $f(t,\theta)$,~ $t>0$,~
$\theta\in \mathbb R/2\pi\mathbb Z$. ~ Here $\theta$  is the angle
variable on $\mathbb S^1$, and  $f$ is $2\pi$-periodic function of
$\theta$. Recall that $s$ is arc-length variable on the curve
$\Gamma(t)=f(t, \mathbb S^1)$ associated with $f$. We have
\begin{equation}\label{appb2}\begin{split}
\partial_s =\frac{1}{|\partial_\theta f|}\, \partial_\theta, \quad
\nabla_s= \partial_s\cdot-(\tau\cdot\partial_s\cdot)\,\tau,\\
\tau=\partial_s f, \quad k=\partial_s^2 f=\partial_s\tau.
\end{split}\end{equation}
Note  that the time derivative $\partial_t$ is evaluated for fixed
angle variable $\theta$. With this notation $\partial_s$ and
$\nabla_s$ become nonlinear differential operators.

Furthermore, we denote by $\mathcal L(t)$ the length
\begin{equation}\label{appb3}
\mathcal
L(t)=\int_{\Gamma(t)}ds:=\int_0^{2\pi}\sqrt{g(t,\theta)}\,d\theta,
\quad \sqrt{g}=|\partial_\theta f|,
\end{equation}
of $\Gamma(t)=f(t, \mathbb S^1)$.





 Now fix an arbitrary even integer
$m\geq 6$. Throughout of the section we  assume that all conditions
of Theorem \ref{theoremsobolev} are satisfied. In particular, there
are two positive constants $E_0$ and $E_m$ such that the initial
curvature $k_0=\partial_s^2 f_0(s)$ satisfies the inequalities
\begin{equation}\label{appb4}
\int_{\Gamma(0)}(\frac{1}{2}|k_0|^2+1)ds\leq E_0, \quad
\int_{\Gamma(0)}|\partial_s^{m-2}k_0|^2\,ds\leq E_m.
\end{equation} The
proof of Theorem \ref{theoremsobolev} falls into  six steps. Our
first task is introduction of notation and presentation of auxiliary
material.
\subsection{Commutators and connections}


\subsubsection{Commutators}




 In the  proofs of a priori estimates in Theorem \ref{theoremsobolev}
 the multiplication of
both sides of  \eqref{appb1} by the higher order  derivatives of $k$
followed by integration  by parts is performed.  This procedure
requires the calculation of the commutator of spatial and time
derivatives. In order to do this, we introduce additional notation.
Let us consider a one-parametric family of immersions $f:[0,T]\times
\mathbb S^1\to \mathbb R^2$. Set
\begin{equation}\label{appb15}
    V=\partial_t\, f, \quad \pi_2= 2V\cdot k, \quad
    \pi_1=\frac{1}{2}\partial_s\pi_2.
\end{equation}
Denote by $\Pi$ a projection and by $\nabla_t$ a differential
operator:
\begin{equation}\label{appb16}
    \Pi\,\Phi=\Phi-(\tau\cdot \Phi)\, \tau, \quad
    \nabla_t\Phi=\Pi\,\partial_t\Phi,
\end{equation}
where $\Phi:[0,T]\times \mathbb S^1:\to \mathbb R^2$ is an arbitrary
smooth vector field. In particular, we have
\begin{equation*}
\nabla_s\Phi= \partial_s\Phi-(\tau\cdot\partial_s\Phi)\, \tau =
\Pi\,\partial_s\Phi, \quad \nabla_t\Phi=
\partial_t\Phi-(\tau\cdot\partial_t\Phi)\, \tau = \Pi\,\partial_t\Phi.
\end{equation*}
The operator $\partial_s^2 \,\partial_t$ takes the following form.

\begin{lemma}\label{appb17}
Under the above assumptions, we have
\begin{equation}\label{appb18}
    \Pi\partial_s^2\partial_t\, f=\nabla_t k-\pi_2\, k.
\end{equation}
\end{lemma}
\begin{proof}
Let $\Phi:[0,T]\times\mathbb S^1:\to \mathbb R^2$ be an arbitrary
smooth vector field. We have
\begin{equation*}\begin{split}
    (\partial_s\partial_t-\partial_t\partial_s
    )\Phi=\frac{1}{|\partial_\theta f|}\partial_\theta\,\partial_t\Phi
    -\partial_t\Big(\frac{1}{|\partial_\theta f|}\,\partial_\theta
    \Phi\Big)=\\
    -\partial_t\Big(\frac{1}{|\partial_\theta f|}\Big)\,
    \partial_\theta\, \Phi=\frac{1}{|\partial_\theta f|^3}\,
    (\partial_\theta f\cdot \partial_\theta\partial_t\,f)\,
    \partial_\theta\Phi=\\
    (\partial_s\, f\cdot \partial_s\partial_t\, f)\, \partial_s\Phi=
    -(\partial_s^2 f\cdot \partial_t
    f)\,\partial_s\Phi=-\frac{1}{2}\pi_2\, \partial_s\Phi.
\end{split}\end{equation*}
Here we use the relation $\partial_sf\cdot\partial_t
f=\tau\cdot\partial_t f=0$, which follows from equation
\eqref{appb1}. Thus we get
\begin{equation}\label{appb19}
    \partial_s\partial_t\Phi=\partial_t\partial_s\Phi-\frac{1}{2}\pi_2\,\partial_s\Phi.
\end{equation}
Next, we have
 \begin{equation*}\begin{split}
\partial_s^2\partial_t \Phi=
\partial_s(\partial_s\partial_t\Phi-\partial_t\partial_s\Phi)+
(\partial_s\partial_t(\partial_s\Phi)-\partial_t\partial_s(\partial_s\Phi)+
\partial_t\partial_s^2\Phi.
\end{split}\end{equation*}
From this and \eqref{appb19} we obtain
 \begin{equation*}\begin{split}
\partial_s^2\partial_t
\Phi=\partial_t\partial_s^2\Phi-\frac{1}{2}\partial_s(\pi_2\partial_s\Phi)
-\frac{1}{2}\pi_2\partial_s^2\Phi.
\end{split}\end{equation*}
Substituting $\Phi=f$ in this relation and recalling the equalities
$\tau=\partial_s f$, $k=\partial_s^2 f$ we arrive at the identity
 \begin{equation*}\begin{split}
\partial_s^2\partial_t
f=\partial_t k-\frac{1}{2}\partial_s(\pi_2)\tau -\pi_2 k.
\end{split}\end{equation*}
Applying to both sides of this identity the projection $\Pi$ and
noting that
$$
\Pi \, \partial_t=\nabla_t, \quad \Pi\, \tau=0, \quad \Pi\, k=k
$$
we obtain the desired equality \eqref{appb18}.

\end{proof}


Now we evaluate the commutator of the differential operators
$\nabla_s^2$ and $\nabla_t$. The result is given by the following
lemma.

\begin{lemma}\label{appb20}
Let $\Phi:[0,T]\times \mathbb S^1$ be a smooth normal field, i.e.,
$\Phi$ is orthogonal to the tangent vector $\tau$. Then we have
\begin{equation}\label{appb21}
    \nabla_s^2\nabla_t
    \Phi=\nabla_t\nabla_s^2\Phi-\pi_2\nabla_s^2\Phi-\pi_1\nabla_s\Phi.
\end{equation}
\end{lemma}

\begin{proof}
Following Lemma 2.1 in \cite{AP}, Lemma 2.1 in \cite{DKS}, and Lemma
1 in \cite{L} we have the identity
\begin{equation} \label{appb22}
\nabla_s^2\nabla_t
    \Phi=\nabla_t\nabla_s^2\Phi-\mathcal C_2-\mathcal C_1-\mathcal
    C_0,
\end{equation}
where $\mathcal C_2=\pi_2\nabla_s^2\Phi$ and
\begin{equation*}\begin{split}
 \mathcal C_1=(k\cdot\nabla_s V+\nabla_s k\cdot V)\nabla_s\Phi
+2\big(\, (k\cdot \nabla_s\Phi)\nabla_s V-(\nabla_s
V\cdot\nabla_s\Phi) k\,\big),\\
\mathcal C_0=\big(\, (\nabla_s k\cdot\Phi)\nabla_s V-(\nabla_s
V\cdot\Phi)\nabla_s k\,\big)+\big(\,( k\cdot\Phi)\nabla_s^2
V-(\nabla_s^2 V\cdot\Phi) k\,\big).
\end{split}\end{equation*}
Note that $\mathcal C_i$ are orthogonal to $\tau$ since $\nabla_s$
and $k$ are parallel to $n$. It is easy to see that
$$
\big(\nabla_s k\cdot\Phi)\nabla_s V-(\nabla_s V\cdot\Phi)\nabla_s
k\,\big)\cdot \Phi=0.
$$
Note that the multipliers in this equality are parallel to $n$.
Hence, either $(\nabla_s k\cdot\Phi)\nabla_s V-(\nabla_s
V\cdot\Phi)\nabla_sk=0$ or $\Phi$ is orthogonal to the normal vector
$n$. In the latter case $\Phi$ is orthogonal to $n$ and $\tau$ and
hence $\Phi=0$. From this we conclude that
$$
(\nabla_s k\cdot\Phi)\nabla_s V-(\nabla_s V\cdot\Phi)\nabla_s k=0.
$$
Next we have
$$
\big(\,( k\cdot\Phi)\nabla_s^2 V-(\nabla_s^2 V\cdot\Phi)
k\,\big)\cdot\Phi=0.
$$
Arguing as before we conclude that either $( k\cdot\Phi)\nabla_s^2
V-(\nabla_s^2 V\cdot\Phi) k=0$ or $\Phi=0$. We thus get
$$
( k\cdot\Phi)\nabla_s^2 V-(\nabla_s^2 V\cdot\Phi) k=0.
$$
Combining the obtained results we conclude that $\mathcal C_0=0$.
Now consider the quantity $\mathcal C_1$. We have
$$
\big(\, (k\cdot \nabla_s\Phi)\nabla_s V-(\nabla_s
V\cdot\nabla_s\Phi) k\,\big)\cdot\nabla_s\Phi=0.
$$
Repeating the previous arguments we conclude that either $(k\cdot
\nabla_s\Phi)\nabla_s V-(\nabla_s V\cdot\nabla_s\Phi) k=0$ or
$\nabla_s\Phi=0$ is orthogonal to $n$ and $\tau$. Hence
$$
(k\cdot \nabla_s\Phi)\nabla_s V-(\nabla_s V\cdot\nabla_s\Phi) k=0
$$
and
$$
\mathcal C_1=(k\cdot\nabla_s V+\nabla_s k\cdot V)\nabla_s\Phi.
$$
Since $V$ and $k$ are orthogonal to $\tau$, we have
$$
k\cdot\nabla_s V+\nabla_s k\cdot V=k\cdot\partial_s V+\partial_s
k\cdot V=\partial_s(k\cdot V)=\pi_1,
$$
which yields the equality $\mathcal C_1=\pi_1\nabla_s\Phi$.
Combining this result with the equalities $\mathcal
C_2=\pi_2\nabla_s^2\Phi$, $\mathcal C_0=0$, and relation
\eqref{appb22} we obtain the desired identity \eqref{appb21}.
\end{proof}


\subsubsection{Relation between $\nabla_s$ and $\partial_s$. Estimates of
$\pi_i$.}  The important ingredient of the theory are estimates
$H^r_\sharp$- Sobolev's norm of the curvature $k$ via $L^2$-norm of
the connection $\nabla_s^r k$. In this section we consider this
problem in many details.  We begin with the observation that every
smooth normal vector field $\Phi:\mathbb S^1\to \mathbb R^2$ admits
the representation
\begin{equation}\label{appb36}
    \Phi=\varphi\, n,\quad \varphi=\Phi\cdot n.
\end{equation}
It is easily seen that for every integer $r>0$ we have
\begin{equation}\label{appb37}
    \nabla_s^r \Phi=\partial_s^r\varphi\, n.
\end{equation}
The following lemma is the main result of this section.
\begin{lemma}\label{appb38}
Let $\Phi:\mathbb S^1\to \mathbb R^2$ be an arbitrary smooth vector
field and $\sigma\in (1/2, 1)$. Then for every integer $r>0$, there
is a constant $c$, depending only on $\sigma$, $r$, and $E_0$, such
that
\begin{equation}\label{appb39}
\|\partial_s^r\Phi-\nabla_s^r\Phi\|_{H^0_\sharp}\leq\\
c\big(\|\Phi\|_{H^0_\sharp}+\|k\|_{H^0_\sharp}\big)^{1+\frac{1-\sigma}{r}}\,
\big(\|\Phi\|_{H^r_\sharp}+\|k\|_{H^r_\sharp}\big)^{1-\frac{1-\sigma}{r}}.
\end{equation}
\end{lemma}
\begin{proof} The proof is given in Section \ref{APPE}.\end{proof}
If we assume that $L^2$-norms ($H^0_\sharp$ -norms) of $\Phi$ and
$k$ are uniformly bounded, then Lemma \ref{appb38} leads to  the
efficient estimate of the deviation $\partial_s\Phi-\nabla_s\Phi$.
The corresponding result is given by the following

\begin{lemma}\label{appb50}
Let under the assumptions of Lemma \ref{appb38},
\begin{equation*}
    \|\Phi\|_{H^0_\sharp}\leq C_\Phi, \quad
    \|k\|_{H^0_\sharp}\leq E_0.
\end{equation*}
Then for every $\epsilon\in (0,1)$, there is a constant $C$,
depending on $E_0$, $C_\Phi$, integer $r\geq 1$, and $\epsilon$,
such that
\begin{equation}\label{appb52}
(1-\epsilon)\|\Phi\|_{H^r_\sharp}-\epsilon\|k\|_{H^r_\sharp}-C\leq
    \|\nabla_s^r \Phi\|_{L^2(0,\mathcal L)}\leq
    (1+\epsilon)\|\Phi\|_{H^r_\sharp}+\epsilon\|k\|_{H^r_\sharp}+C.
\end{equation}
\end{lemma}
\begin{proof} Estimate \eqref{appb39} and the
conditions of  lemma imply the inequality
\begin{equation*}
\|\partial_s^r\Phi-\nabla_s^r\Phi\|_{H^0_\sharp}\leq C
\big(\|\Phi\|_{H^r_\sharp}+\|k\|_{H^r_\sharp}\big)^{1-\frac{1-\sigma}{r}}.
\end{equation*}
Since $1/2<\sigma<1$ it follows from this and the Young inequality
that
\begin{equation*}
\|\partial_s^r\Phi-\nabla_s^r\Phi\|_{H^0_\sharp}\leq C+\epsilon
\big(\|\Phi\|_{H^r_\sharp}+\|k\|_{H^r_\sharp}\big).
\end{equation*}
We thus get the inequality
\begin{equation*}
\big|\,
\|\partial_s^r\Phi\|_{H^0_\sharp}-\|\nabla_s^r\Phi\|_{H^0_\sharp}
\,\big|\leq C+\epsilon
\big(\|\Phi\|_{H^r_\sharp}+\|k\|_{H^r_\sharp}\big).
\end{equation*}
which yields
\begin{equation*}\begin{split}
\|\partial_s^r\Phi\|_{H^0_\sharp}-\epsilon
\big(\|\Phi\|_{H^r_\sharp}+\|k\|_{H^r_\sharp}\big)-C\leq
\|\nabla_s^r\Phi\|_{H^0_\sharp}\\ \leq
\|\partial_s^r\Phi\|_{H^0_\sharp}+\epsilon
\big(\|\Phi\|_{H^r_\sharp}+\|k\|_{H^r_\sharp}\big)+C.
\end{split}
\end{equation*}
Noting that
$$
\|\Phi\|_{H^r_\sharp}- C_\Phi\leq \|\partial_s^r\Phi\|_{L^2(0,
\mathcal L)} \leq \|\Phi\|_{H^r_\sharp}+ C_\Phi,
$$
we arrive at desired estimate \eqref{appb52}.
\end{proof}

\begin{corollary}\label{appb53}
Let $\|k\|_{L^2(0,\mathcal L)}\leq \sqrt{2E_0}$ and an integer
$r\geq 1$. Then for every $\epsilon\in (0,1)$ there exists a
constant $C$, depending only on $E_0$, $r$, $\epsilon$, such that
\begin{equation}\label{appb54}
(1-2\epsilon)\|k\|_{H^r_\sharp}-C\leq
    \|\nabla_s^r k\|_{H^0_\sharp}\leq
    (1+2\epsilon)\|k\|_{H^r_\sharp}+C.
\end{equation}
\end{corollary}
\begin{proof}It suffices to note that $\Phi=k$ and $C_\Phi=\sqrt{2E_0}$ meet all
requirements of Lemma \ref{appb50}.
\end{proof}


\paragraph{ Estimates of $\pi_i$.} In this paragraph we give estimates for
the coefficients $\pi_i$ defined  by \eqref{appb15}. The results are
given by the following Lemmas.
\begin{lemma}\label{appb61}
Under the assumptions of Theorem  \ref{theoremsobolev} for every
integer $r\geq 0$, there is a constant $c$ depending on $r$, such
that
\begin{equation}\label{appb62}
    \|V(t)\|_{H^r_\sharp}\leq c (1+\|k(t)\|_{H^{r+2}_\sharp}).
\end{equation}
\end{lemma}

\begin{proof} Notice that
\begin{equation}\label{appb61a}
V\equiv \partial_t f=-\nabla_s^2 k\,-\,\frac{1}{2}|k|^2
k\,+\,k\,-dJ.
\end{equation}
By virtue of Corollary \ref{appb53}
\begin{equation}\label{appb62a}
\|\nabla_s^2k\|_{H^r_\sharp}\leq c+
c\|\nabla_s^{r+2}k\|_{H^0_\sharp}\leq c+c\|k\|_{H^{r+2}_\sharp}.
\end{equation}
Next, the Moser inequality \eqref{appb11} implies the estimate
\begin{equation*}
    \||k|^2 k\|_{H^r_\sharp}\leq c\|k\|_{L^\infty(0,\mathcal L)}^2\,
    \|k\|_{H^r_\sharp}.
\end{equation*}
Now choose an arbitrary $\sigma\in (1/2,1)$. Recall that the
embedding $H^\sigma_\sharp\hookrightarrow L^\infty(0,\mathcal L)$ is
continuous. From this, the interpolation inequality, and the
estimate $\|k\|_{L^2(0,\mathcal L)}\leq c$ we obtain
\begin{equation}\begin{split}\label{appb63}
    \||k|^2 k\|_{H^r_\sharp}\leq c\,\,\|k\|_{H^\sigma_\sharp}^2\,\,
    \|k\|_{H^r_\sharp}\leq
    c\,\,\|k\|_{H^{r+2}_\sharp}^{\frac{2\sigma}{r+2}}\,\,\,
    \|k\|_{H^{r+2}_\sharp}^{\frac{r}{r+2}}=\\
   c\,\,\|k\|_{H^{r+2}_\sharp}^{\frac{r+2\sigma}{r+2}} \leq c\,\,(1+\|k\|_{H^{r+2}_\sharp}).
\end{split}\end{equation}
Finally, Theorem \ref{bokr60} on the estimates of the Kohn-Vogelius
functional and estimate \eqref{appb54} in Corollary \ref{appb53}
imply
\begin{equation}\label{appb64}
\|dJ\|_{H^r_\sharp}\leq  c+c\|k\|_{H^r_\sharp}.
\end{equation}
Combining estimates \eqref{appb62a} - \eqref{appb64} and recalling
relation \eqref{appb61a} we arrive at desired estimate
\eqref{appb62}.
\end{proof}

We are now in a position to estimate the quantities $\pi_i$. The
result is given by the following lemma.

\begin{lemma}\label{appb65}
Assume that all assumptions of Theorem \ref{theoremsobolev} are
satisfied. Let an integer $r\geq 0$ and $\sigma\in (1/2,1)$ be
given. Then there is $c$, depending on $r$, $\sigma$ and the
constants $\nu,$ $\varrho$ in Theorem \ref{theoremsobolev}, such
that
\begin{equation}\label{appb66}
\|\pi_2(t)\|_{H^r_\sharp}\leq
c(1+\|k(t)\|_{H^{r+2}_\sharp}^{1+\frac{\sigma}{r+2}}), \quad
\|\pi_1(t)\|_{H^r_\sharp}\leq
c(1+\|k(t)\|_{H^{r+3}_\sharp}^{1+\frac{\sigma}{r+3}})
\end{equation}
for all $t\in (0,T)$.
\end{lemma}

\begin{proof} Since $\pi_1=-\partial_s\pi_2/2$, it suffices to
estimate $\pi_2$. To shorten notation,  we omit the symbol $t$.
Notice that $\pi_2=k\cdot V$. From this,  the Moser inequality, and
continuity of the embedding $H^\sigma_\sharp\hookrightarrow
L^\infty(0, \mathcal L)$ we conclude that
\begin{equation}\label{appb67}
    \|\pi_2\|_{H^r_\sharp}\leq
    c\|k\|_{H^\sigma_\sharp}\|V\|_{H^r_\sharp}+\|k\|_{H^r_\sharp}\|V\|_{H^\sigma_\sharp}.
\end{equation}
By the interpolation inequality and estimate \eqref{appb62} in Lemma
\ref{appb61}, we have
\begin{equation*}
\|V\|_{H^\sigma}\leq \|V\|_{H^0_\sharp}^{1-\frac{\sigma}{r}}\,
\|V\|_{H^r_\sharp}^{\frac{\sigma}{r}}\leq
c(1+\|k\|_{H^{2}_\sharp})^{1-\frac{\sigma}{r}}\,\,(1+\|k\|_{H^{r+2}_\sharp})^{\frac{\sigma}{r}}.
\end{equation*}
Since
$$
\|k\|_{H^2_\sharp}\leq
\|k\|_{H^0_\sharp}^{1-\frac{2}{r+2}}\,\,\,\|k\|_{H^{r+2}_\sharp}^{\frac{2}{r+2}}\leq
c \,\,\|k\|_{H^{r+2}_\sharp}^{\frac{2}{r+2}},
$$
we have
\begin{equation*}\begin{split}
\|V\|_{H^\sigma_\sharp}\leq
c\,\big(\,(1+\|k\|_{H^{r+2}_\sharp}^{\frac{2}{r+2}}\,\big)^{1-\frac{\sigma}{r}}\,\,
(1+\|k\|_{H^{r+2}_\sharp})^{\frac{\sigma}{r}}\leq
c(1+\|k\|_{H^{r+2}_\sharp})^\mu,
\end{split}\end{equation*}
where
$$
\mu=\frac{2}{r+2}(1-\frac{\sigma}{r})+\frac{\sigma}{r}=\frac{2+\sigma}{r+2}.
$$
From this and the inequality
$$
\|k\|_{H^r_\sharp}\leq c \,
\|k\|_{H^0_\sharp}^{1-\frac{r}{r+2}}\,\,\|k\|_{H^{r+2}_\sharp}^{\frac{r}{r+2}}\leq
c \|k\|_{H^{r+2}_\sharp}^{\frac{r}{r+2}}
$$
we obtain
\begin{equation}\label{appb68}
    \|k\|_{H^r_\sharp}\|V\|_{H^\sigma_\sharp}\leq
    c\,\|k\|_{H^{r+2}_\sharp}^{\frac{r}{r+2}}\,\,\,(1+\|k\|_{H^{r+2}_\sharp})^\mu\leq
    c+c\|k\|_{H^{r+2}_\sharp}^{1+\frac{\sigma}{r+2}}.
\end{equation}
Next, the interpolation inequality implies
$$
\|k\|_{H^\sigma_\sharp}\leq c\,\,
\|k\|_{H^0_\sharp}^{1-\frac{\sigma}{r+2}}\,\,\|k\|_{H^{r+2}_\sharp}^{\frac{\sigma}{r+2}}\leq
c\, \|k\|_{H^{r+2}_\sharp}^{\frac{\sigma}{r+2}},
$$
which along with estimate \eqref{appb62} in Lemma \ref{appb61} gives
\begin{equation*}
    \|k\|_{H^\sigma_\sharp}\|V\|_{H^r_\sharp}\leq
    c\,\|k\|_{H^{r+2}_\sharp}^{\frac{\sigma}{r+2}}\, \,(1+\|k\|_{H^{r+2}_\sharp})\leq
    c+c\,\|k\|_{H^{r+2}_\sharp}^{1+\frac{\sigma}{r+2}}.
\end{equation*}
Substituting this estimate and estimate \eqref{appb68} into
\eqref{appb67} we arrive at desired estimate \eqref{appb66}.
\end{proof}

























\subsection{Main integral identity.} Now we use  formulae
\eqref{appb18} and \eqref{appb21}  in order to derive the main
integral identity for solutions to problem \eqref{appb1}. The result
is given by the following proposition.
\begin{proposition}\label{appb23} The following integral identity holds
for every smooth solution $f:[0,T]\times \mathbb S^1$ to problem
\eqref{appb1}, for every~ $t_0\in (0,T]$, and for every even integer
$m\geq 6$.
\begin{equation}\label{appb24}\begin{split}
\frac{1}{2}\int_{\Gamma(t_0)}|\nabla_s^{m-2} k(t_0)|^2\,ds
+\int_0^{t_0}\int_{\Gamma(t)}|\nabla_s^{m} k(t)|^2\,dsdt=\\\sum_1^4
\mathcal N_i+\frac{1}{2}\int_{\Gamma(0)}|\nabla_s^{m-2} k(0)|^2\,ds.
\end{split}\end{equation}
Here the quantity $\mathcal N_i$ are defined by the equalities
\begin{equation}\label{appb24e}\begin{split}
    \mathcal N_1=\sum_{i=0}^{\frac{m-2}{2}}\int_0^{t_0}I_{2i}(t)\,dt
    +\frac{1}{2}I_{m-2},\,\,\,\,
  \mathcal N_2=  \sum_{i=1}^{\frac{m-2}{2}}
    \int_0^{t_0}I_{2i-1}(t)\,dt,\\
I_{2i}(t)=\int_{\Gamma(t)}\nabla_s^{m-2-2i}
\big(\,\pi_2\nabla_s^{2i}k\,\big)\cdot \nabla_s^{m-2}k\,ds,\\
I_{2i-1}(t)=\int_{\Gamma(t)}\nabla_s^{m-2-2i}
\big(\,\pi_1\nabla_s^{2i-1}k\,\big)\cdot \nabla_s^{m-2}k\,ds,
\end{split}\end{equation}
\begin{equation}\label{appb24f}\begin{split}
\mathcal N_3=\int_0^{t_0}\int_{\Gamma(t)}\nabla_s^{m-4}\big(\, (
k\cdot\nabla_s^2 k)\,k\big)\cdot\nabla_s^m k\, dsdt
,\\
\mathcal N_4=-\int_0^{t_0}\int_{\Gamma(t)}
\nabla_s^{m-2}\Upsilon\cdot\nabla_s^m
k\,dsdt+\int_0^{t_0}\int_{\Gamma(t)}\nabla_s^{m-4}\big((k\cdot
\Upsilon)\,k)
    \cdot\nabla_s^{m}k\,dsdt,
\end{split}\end{equation}
where
$$
\Upsilon=\frac{1}{2}|k|^2\, k\,-\, k\,+\, dJ.
$$
\end{proposition}

\begin{proof} Multiplying both sides of \eqref{appb1} by
$\partial_s^2\nabla_s^{2m-4}k$ and integrating the result with
respect to $s$ and  $t$ we arrive at the equality
\begin{equation}\label{appb25}\begin{split}
    \int_0^{t_0}\int_{\Gamma(t)}\Big(\partial_t
    f\cdot\partial_s^2\nabla_s^{2m-4}k
+\nabla_s^2 k\cdot\partial_s^2\nabla_s^{2m-4}k\Big)\, dsdt+\\
\int_0^{t_0}\int_{\Gamma(t)}\Big(\frac{1}{2}|k|^2k-k+dJ\Big)\cdot
\partial_s^2\nabla_s^{2m-4}k\,dsdt=0.
\end{split}\end{equation}
The rest of the proof falls into a sequence of lemmas.

\begin{lemma}\label{appb25a}
Under the assumptions of Proposition \ref{appb23}, we have
\begin{equation}\label{appb24a}\begin{split}
\int_0^{t_0}\int_{\Gamma(t)}\partial_t
    f\cdot\partial_s^2\nabla_s^{2m-4}k\, ds=
    \frac{1}{2}\int_{\Gamma(t_0)} |\nabla_s^{m-2}\,k(t_0)|^2\, ds\\
-\frac{1}{2}\int_{\Gamma(0)} |\nabla_s^{m-2}\,k(0)|^2\, d s-\mathcal
N_1-\mathcal N_2. \end{split}\end{equation}

\end{lemma}

\begin{proof} Notice that $\nabla_s^\top=-\partial_s\Pi$ and
$\nabla_s^{m-2}k=\Pi\nabla_s^{m-2}k$. Since $m$ is even, it follows
that
\begin{equation*}\begin{split}
\int_{\Gamma(t)}\partial_t
    f\cdot\partial_s^2\nabla_s^{2m-4}k\,ds=
\int_{\Gamma(t)}(\nabla_s^\top)^{m-2}\partial_s^2\partial_t\,f\cdot
\Pi\,\nabla_s^{m-2} k\,ds=\\
\int_{\Gamma(t)}\nabla_s^{m-2}\Pi\partial_s^2\partial_t\,f\cdot
\nabla_s^{m-2} k\,ds.
\end{split}\end{equation*}
Recall that the integer $m$ is even. From this and identity
\eqref{appb18} in Lemma \ref{appb17} we obtain
\begin{equation*}\begin{split}
\int_{\Gamma(t)}\partial_t
    f\cdot\partial_s^2\nabla_s^{2m-4}k\,ds=
 \int_{\Gamma(t)}\nabla_s^{m-2}\partial_t\,k\cdot
\nabla_s^{m-2} k\,ds- \int_{\Gamma(t)}\nabla_s^{m-2}(\pi_2\,k)\cdot
\nabla_s^{m-2} k\,ds.
\end{split}\end{equation*}
Recalling the expression for $I_{2i}(t)$ we can rewrite this
equality in the equivalent form
\begin{equation}\begin{split}\label{appb26}
\int_{\Gamma(t)}\partial_t
    f\cdot\partial_s^2\nabla_s^{2m-4}k\,ds=
 \int_{\Gamma(t)}\nabla_s^{m-2}\nabla_t\,k\cdot
\nabla_s^{m-2} k\,ds- I_0(t).
\end{split}\end{equation}
Let us consider the  integral in the right hand side of this
equality.  Using identity \eqref{appb21} in Lemma \ref{appb20} with
$\Phi$ replaced by $k$ we obtain
\begin{equation*}\begin{split}
 \int_{\Gamma(t)}\nabla_s^{m-2}\nabla_t\,k\cdot
\nabla_s^{m-2}
k\,ds=\int_{\Gamma(t)}\nabla_s^{m-4}\nabla_t\nabla_s^2\,k\cdot
\nabla_s^{m-2} k\,ds-\\
\int_{\Gamma(t)}\nabla_s^{m-4}(\pi_2\nabla_s^2\,k)\cdot
\nabla_s^{m-2}
k\,ds-\int_{\Gamma(t)}\nabla_s^{m-4}(\pi_1\nabla_s\,k)\cdot
\nabla_s^{m-2} k\,ds
\end{split}\end{equation*}
or equivalently
\begin{equation*}\begin{split}
 \int_{\Gamma(t)}\nabla_s^{m-2}\nabla_t\,k\cdot
\nabla_s^{m-2}
k\,ds=\int_{\Gamma(t)}\nabla_s^{m-4}\nabla_t\nabla_s^2\,k\cdot
\nabla_s^{m-2} k\,ds-I_2(t)-I_1(t).
\end{split}\end{equation*}
Repeating this process and using again identity \eqref{appb21} we
finally obtain
\begin{equation*}\begin{split}
 \int_{\Gamma(t)}\nabla_s^{m-2}\nabla_t\,k\cdot
\nabla_s^{m-2} k\,ds=\int_{\Gamma(t)}\nabla_t\nabla_s^{m-2}\,k\cdot
\nabla_s^{m-2}
k\,ds-\sum_{i=1}^{\frac{m-2}{2}}(I_{2i}(t)-I_{2i-1}(t)).
\end{split}\end{equation*}
Notice that
$$
\nabla_t\nabla_s^{m-2}\,k\cdot \nabla_s^{m-2}
k=\partial_t\nabla_s^{m-2}\,k\cdot \nabla_s^{m-2} k
$$
since $\nabla_s^{m-2} k$ is orthogonal to $\tau$. Thus we get
\begin{equation*}\begin{split}
 \int_{\Gamma(t)}\nabla_s^{m-2}\nabla_t\,k\cdot
\nabla_s^{m-2}
k\,ds=\int_{\Gamma(t)}\partial_t\nabla_s^{m-2}\,k\cdot
\nabla_s^{m-2}
k\,ds-\sum_{i=1}^{\frac{m-2}{2}}(I_{2i}(t)-I_{2i-1}(t)).
\end{split}\end{equation*}
Combining this equality with \eqref{appb26} we arrive at the
identity
\begin{equation}\label{appb27}\begin{split}
\int_{\Gamma(t)}\partial_t
    f\cdot\partial_s^2\nabla_s^{2m-4}k\,ds=
    \int_{\Gamma(t)}\partial_t\nabla_s^{m-2}\,k\cdot
\nabla_s^{m-2}
k\,ds\\-\sum_{i=0}^{\frac{m-2}{2}}I_{2i}(t)-\sum_{i=1}^{\frac{m-2}{2}}
I_{2i-1}(t)).
\end{split}\end{equation}
Now consider the integral
$$
\int_{0}^{t_0}\int_{\Gamma(t)}\partial_t\nabla_s^{m-2}\,k\cdot
\nabla_s^{m-2} k\,dsdt.
$$
Our task is to integrate by parts with respect to $t$. The
difficulty is that the curve $\Gamma(t)$ and $ds$ depend on the time
variable. In order to cope with this difficulty we rewrite this
integral in terms of the original independent variable $\theta$. The
change of variable $s\to \theta$ leads to the equality
$$
\int_{0}^{t_0}\int_{\Gamma(t)}\partial_t\nabla_s^{m-2}\,k\cdot
\nabla_s^{m-2} k\,dsdt=\int_0^{t_0}\int_0^{2\pi}
\partial_t\nabla_s^{m-2}\,k\cdot \nabla_s^{m-2} k\,\sqrt{g}\,
d\theta dt,
$$
where $\sqrt{g}=|\partial_\theta f|$. Integrating by parts gives the
equality
\begin{equation}\label{appb28}\begin{split}
\int_0^{t_0}\int_0^{2\pi}
\partial_t\nabla_s^{m-2}\,k\cdot \nabla_s^{m-2} k\,\sqrt{g}\,
d\theta dt=\\\frac{1}{2}\int_0^{2\pi}
|\nabla_s^{m-2}\,k(t_0)|^2\,\sqrt{g(t_0)}\, d\theta
-\frac{1}{2}\int_0^{2\pi} |\nabla_s^{m-2}\,k(0)|^2\,\sqrt{ g(0)}\,
d\theta\\ -\frac{1}{2}\int_0^{t_0}\int_0^{2\pi}
|\nabla_s^{m-2}\,k(t)|^2\,\partial_t\sqrt{g(t)}\, d\theta dt.
\end{split}\end{equation}
Next we have
$$
\partial_t\sqrt{g(t)}=\frac{1}{|\partial_\theta f|}
(\partial_\theta f\cdot \partial_\theta\partial_t f) = (\partial_s
f\cdot\partial_s\partial_t f)\sqrt{g(t)}.
$$
Notice that
$$
\partial_s
f\cdot\partial_s\partial_t f=\tau\cdot\partial_s\partial_t f
=-\partial_s\tau\cdot \partial_t f=-k\cdot V=-\frac{1}{2}\pi_2,
$$
which yields
\begin{equation}\begin{split}\label{appb29}
\int_0^{2\pi} |\nabla_s^{m-2}\,k(t)|^2\,\partial_t\sqrt{\mathbf
g(t)}\, d\theta=-\frac{1}{2}\int_0^{2\pi}
|\nabla_s^{m-2}\,k(t)|^2\,\pi_2\sqrt{g(t)}\, d\theta =\\
-\frac{1}{2}\int_{\Gamma(t)}|\nabla_s^{m-2}\,k(t)|^2\,\pi_2\, ds=
-\frac{1}{2}I_{m-2}(t).
\end{split}\end{equation}
Next, we have
\begin{equation*}\begin{split}
\frac{1}{2}\int_0^{2\pi} |\nabla_s^{m-2}\,k(t_0)|^2\,\sqrt{\mathbf
g(t_0)}\, d\theta -\frac{1}{2}\int_{0}^{2\pi}
|\nabla_s^{m-2}\,k(0)|^2\,\sqrt{g(0)}\, d\theta=\\
\frac{1}{2}\int_{\Gamma(t_0)} |\nabla_s^{m-2}\,k(t_0)|^2\, ds
-\frac{1}{2}\int_{\Gamma(0)} |\nabla_s^{m-2}\,k(0)|^2\, d s.
\end{split}\end{equation*}
Substituting this equality along with equality \eqref{appb29} into
\eqref{appb28} we obtain
\begin{equation}\begin{split}\label{appb30}
\int_{0}^{t_0}\int_{\Gamma(t)}\partial_t\nabla_s^{m-2}\,k\cdot
\nabla_s^{m-2} k\,dsdt=\int_0^{t_0}\int_0^{2\pi}
\partial_t\nabla_s^{m-2}\,k\cdot \nabla_s^{m-2} k\,\sqrt{g}\,
d\theta dt\\
= \frac{1}{2}\int_{\Gamma(t_0)} |\nabla_s^{m-2}\,k(t_0)|^2\, ds
-\frac{1}{2}\int_{\Gamma(0)} |\nabla_s^{m-2}\,k(0)|^2\, d s
-\frac{1}{2}\int_0^{t_0} I_{m-2}(t)\, dt.
\end{split}\end{equation}
Integrating both sides of equality \eqref{appb27} and using relation
\eqref{appb30} we arrive at the desired equality \eqref{appb24a}.


\end{proof}

\begin{lemma}\label{appb32}
Under the assumptions of Proposition \ref{appb23} we have
\begin{equation}\label{appb24s}\begin{split}
\int_0^{t_0}\int_{\Gamma(t)}\nabla_s^2k\cdot\partial_s^2\nabla_s^{2m-4}k\,
ds=
    \int_0^{t_0}\int_{\Gamma(t)} |\nabla_s^{m}\,k(t)|^2\, dsdt-
    \mathcal N_3.
\end{split}\end{equation}

\end{lemma}
\begin{proof}Arguing as in the proof of Lemma \ref{appb25a} we
obtain
\begin{equation}\begin{split}\label{appb33}
\int_{\Gamma(t)}\nabla_s^2 k
    \cdot\partial_s^2\nabla_s^{2m-4}k\,ds=
\int_{\Gamma(t)}(\nabla_s^\top)^{m-4}\partial_s^2\nabla_s^2 k\cdot
\Pi\,\nabla_s^{m} k\,ds=\\
\int_{\Gamma(t)}\nabla_s^{m-4}\Pi\,\partial_s^2\,\nabla_s^2\,k\cdot
\nabla_s^{m}
k\,ds=\int_{\Gamma(t)}\nabla_s^{m-4}\,\nabla_s\partial_s\,\nabla_s^2\,k\cdot
\nabla_s^{m} k\,ds.
\end{split}\end{equation}
Note that
\begin{equation*}
    \partial_s\nabla_s^2 k=\nabla_s^3 k+(\tau\cdot
    \partial_s\nabla_s^2k)\, \tau.
\end{equation*}
Since $\nabla_s^2$ is orthogonal to $\tau$, we have
$$
\partial_s\nabla_s^2 k=\nabla_s^3 k-(k\cdot
    \nabla_s^2k)\, \tau.
$$
It is easy to check that
$$
\nabla_s\big((k\cdot
    \nabla_s^2k)\, \tau\,\big)=(k\cdot
    \nabla_s^2k)\, \nabla_s\tau=(k\cdot
    \nabla_s^2k)\,k,
$$
which yields
$$
\nabla_s\partial_s\nabla_s^2 k=\nabla_s^4 k-(k\cdot
    \nabla_s^2k)\,k.
$$
Substituting this relation into \eqref{appb33} we finally obtain
\begin{equation*}\begin{split}
\int_{\Gamma(t)}\nabla_s^2 k
    \cdot\partial_s^2\nabla_s^{2m-4}k\,ds=\\
\int_{\Gamma(t)}| \nabla_s^{m}
k|^2\,ds-\int_{\Gamma(t)}\nabla_s^{m-4}\big((k\cdot\nabla_s^2 k)\,k)
    \cdot\nabla_s^{m}k\,ds,
\end{split}\end{equation*}
which along with the expression for $\mathcal N_3$ implies desired
equality \eqref{appb24a}.
\end{proof}
Recall the denotation
$$
\Upsilon=\frac{1}{2}|k|^2\, k\,-\, k\,+\, dJ.
$$

\begin{lemma}\label{appb34e}
Under the assumptions of Proposition \ref{appb23} we have
\begin{equation}\label{appb34}
    \int_0^{t_0} \Upsilon\cdot\partial_s^2\nabla_s^{2m-4}=
    -\mathcal N_4.
\end{equation}
\end{lemma}

\begin{proof} The proof imitates the proof of Lemma \ref{appb32}. We
have
\begin{equation}\begin{split}\label{appb35}
\int_{\Gamma(t)}\Upsilon
    \cdot\partial_s^2\nabla_s^{2m-4}k\,ds=
\int_{\Gamma(t)}(\nabla_s^\top)^{m-4}\partial_s^2\Upsilon\cdot
\Pi\,\nabla_s^{m} k\,ds\\
=\int_{\Gamma(t)}\nabla_s^{m-4}\,\nabla_s\partial_s\,\Upsilon\cdot
\nabla_s^{m} k\,ds.
\end{split}\end{equation}
Notice that $\Upsilon$ is orthogonal to $\tau$. It follows that
\begin{equation*}
    \partial_s\Upsilon=\nabla_s \Upsilon+(\tau\cdot
    \partial_s\Upsilon)\, \tau=\nabla_s \Upsilon-(k\cdot
    \Upsilon)\, \tau.
\end{equation*}
Next we have
$$
\nabla_s\big((k\cdot
    \Upsilon)\, \tau\,\big)=(k\cdot
    \Upsilon)\, \nabla_s\tau=(k\cdot
    \Upsilon)\,k,
$$
which yields
$$
\nabla_s\partial_s\Upsilon=\nabla_s^2 \Upsilon-(k\cdot
    \Upsilon)\,k.
$$
Substituting this relation into \eqref{appb34} we arrive at the
identity
\begin{equation*}\begin{split}
\int_{\Gamma(t)}\Upsilon
    \cdot\partial_s^2\nabla_s^{2m-4}k\,ds=\\
\int_{\Gamma(t)} \nabla_s^{m-2}\Upsilon\cdot\nabla_s^m
k\,ds-\int_{\Gamma(t)}\nabla_s^{m-4}\big((k\cdot \Upsilon)\,k)
    \cdot\nabla_s^{m}k\,ds.
\end{split}\end{equation*}
which along with the expression for $\mathcal N_4$ yields
\eqref{appb34}.
\end{proof}
We are now in a position to complete the proof of Proposition
\ref{appb23}. To this end, it suffices to substitute equalities
\eqref{appb24a}, \eqref{appb24s}, and \eqref{appb34} into integral
identity \eqref{appb25}.


\end{proof}



\subsection{Estimates of reminders $\mathcal N_i$}




\subsubsection{Estimates of $\mathcal N_1$ and $\mathcal N_2$.}
Estimates \eqref{appb66} for $\pi_i$  lead to the basic estimates
for the reminders $\mathcal N_i$ in the integral identity
\eqref{appb24}. The first result in this direction is given by  the
following
\begin{lemma}\label{appb69}
Assume that all assumptions of Theorem \ref{theoremsobolev} are
satisfied. Let an even integer $m\geq 4$ and $\sigma\in (1/2,1)$ be
given. Then, there is $c$, depending on $m$ and $\sigma$, such that
\begin{equation}\label{appb70}
|\mathcal N_i|\leq c \,\int_0^T
\big(\,1+\|k(t)\|_{H^{m}_\sharp}^{2-\frac{2(1-\sigma)}{m}}\,\big)
\text{~~for ~~} i=1,2 .
\end{equation}
\end{lemma}
\begin{proof}
The proof is based on Lemma \ref{appb65} and Corollary \ref{appb53}.
We give the proof only for the quantity $\mathcal N_1$. The proof
for $\mathcal N_2$ is similar. It follows from representation
\eqref{appb24e} for $\mathcal N_1$ that
\begin{equation}\label{appb71}\begin{split}
    \mathcal N_1=\sum_{i=0}^{\frac{m-2}{2}}\int_0^{t_0}I_{2i}(t)\,dt
    +\frac{1}{2}I_{m-2}(t),\\
 I_{2i}(t)=\int_{\Gamma(t)}\nabla_s^{m-2-2i}
\big(\,\pi_2\nabla_s^{2i}k\,\big)\cdot \nabla_s^{m-2}k\,ds.
\end{split}\end{equation}
Hence it suffices to estimate $I_{2i}(t)$. To simplify the notation,
we omit the symbol $t$. By the Cauchy inequality, we have
\begin{equation}\label{appb72}
|I_{2i}|\leq \|\nabla_s^{m-2-2i}
\big(\,\pi_2\nabla_s^{2i}k\,\big)\|_{H^0_\sharp}\,
\|\nabla_s^{m-2}k\|_{H^0_\sharp}.
\end{equation}
Recall that for every normal vector field $\Phi=\varphi n$, we have
$$
\nabla^r_s\Phi=\partial_s^r\varphi\, n.
$$
It follows that
$$
\pi_2\nabla_s^{2i}k=\pi_2\partial_s^{2i} K\, n, \quad K=k\cdot n
$$
and
$$
\nabla_s^{m-2-2i}
\big(\,\pi_2\nabla_s^{2i}k\,\big)=\partial_s^{m-2-2i}(\pi_2\partial_s^{2i}K)\,
n\,.
$$
Thus we get
\begin{equation}\label{appb73}
\|\nabla_s^{m-2-2i}
\big(\,\pi_2\nabla_s^{2i}k\,\big)\|_{H^0_\sharp}\,\leq
\|\pi_2\partial_s^{2i}K\|_{H^{m-2-2i}_\sharp}.
\end{equation}
The Moser inequality and the continuity of the embedding
$H^\sigma_\sharp\hookrightarrow L^\infty(0, \mathcal L)$ imply the
estimate
\begin{equation}\label{appb74}\begin{split}
\|\pi_2\partial_s^{2i}K\|_{H^{m-2-2i}_\sharp}\leq
c\,\|\pi_2\|_{H^{m-2-2i}_\sharp}\|\partial_s^{2i}K\|_{H^{\sigma}_\sharp}+
c\,\|\pi_2\|_{H^\sigma_\sharp}\|\partial_s^{2i}K\|_{H^{m-2-2i}_\sharp}\\\leq
c\,\|\pi_2\|_{H^{m-2-2i}_\sharp}\|K\|_{H^{2i+\sigma}_\sharp}+
c\,\|\pi_2\|_{H^\sigma_\sharp}\|K\|_{H^{m-2}_\sharp}.
\end{split}\end{equation}
It follows from the interpolation inequality and estimate
\eqref{appb66} in Lemma \ref{appb65} that
\begin{equation}\label{appb75}\begin{split}
 \|\pi_2\|_{H^\sigma_\sharp}\leq
 \|\pi_2\|_{H^0_\sharp}^{1-\frac{\sigma}{m-2}}\,\,
 \|\pi_2\|_{H^{m-2}_\sharp}^{\frac{\sigma}{m-2}}\leq
 \\ c\,
 \big(\,1+\|k\|_{H^2_\sharp}^{1+\frac{\sigma}{2}}\,\big)^{1-\frac{\sigma}{m-2}}\,\,
 \big(\,1+\|k\|_{H^{m}_\sharp}^{1+\frac{\sigma}{m}}\,\big)^{\frac{\sigma}{m-2}}.
\end{split}\end{equation}
Since $\|k\|_{H^0_\sharp}$ is uniformly bounded, the interpolation
inequality implies
$$
\|k\|_{H^2_\sharp}\leq c\|k\|_{H^m_\sharp}^{\frac{2}{m}}.
$$
Substituting this inequality in \eqref{appb75} we arrive at the
estimate
\begin{equation}\label{appb76}\begin{split}
 \|\pi_2\|_{H^\sigma_\sharp}\leq c(1+\|k\|_{H^m_\sharp}^\iota),
\end{split}\end{equation}
where
$$
\iota=\frac{2}{m}(1+\frac{\sigma}{2})(1-\frac{\sigma}{m-2})+
(1+\frac{\sigma}{m})\frac{\sigma}{m-2}=\frac{2+2\sigma}{m}\,.
$$
Next notice that $\|K\|_{H^0_\sharp}=\|k\|_{H^0_\sharp} $ is
uniformly bounded, which yields
$$
\|K\|_{H^{m-2}_\sharp}\leq c\|K\|_{H^m_\sharp}^{\frac{m-2}{m}}\,.
$$
From this and \eqref{appb76} we obtain
\begin{equation}\label{appb77}
\|\pi_2\|_{H^\sigma_\sharp}\,\,\|K\|_{H^{m-2}_\sharp}\leq
c\,(1+\|k\|_{H^m_\sharp})^{\frac{2+2\sigma}{m}}\,\,\|K\|_{H^m_\sharp}^{\frac{m-2}{m}}\,.
\end{equation}
Next,  estimate \eqref{appb66} in Lemma \ref{appb65} and the
interpolation inequality imply
\begin{equation*}\begin{split}
\|\pi_2\|_{H^{m-2-2i}_\sharp}\|K\|_{H^{2i+\sigma}_\sharp}\leq
c\,\big(\,1+\|k\|_{H^{m-2i}_\sharp}^{1+\frac{\sigma}{m-2i}}\,\big)\,\,
\|K\|_{H^m_\sharp}^{\frac{2i+\sigma}{m}}\leq\\
c\,\big(\,1+\|k\|_{H^{m}_\sharp}^{\frac{m-2i}{m}(1+\frac{\sigma}{m-2i})}\,\big)
\,\,\|K\|_{H^m_\sharp}^{\frac{2i+\sigma}{m}},
\end{split}\end{equation*}
which gives
\begin{equation}\begin{split}\label{appb78}
\|\pi_2\|_{H^{m-2-2i}_\sharp}\|K\|_{H^{2i+\sigma}_\sharp}\leq
c(1+\|k\|_{H^{m}_\sharp}^{\frac{m-2i+\sigma}{m}})\,
\|K\|_{H^m_\sharp}^{\frac{2i+\sigma}{m}}.
\end{split}\end{equation}
Substituting \eqref{appb77} and \eqref{appb78} into \eqref{appb74}
and next into \eqref{appb73} we arrive at the inequality
\begin{equation}\label{appb79}\begin{split}
\|\nabla_s^{m-2-2i}
\big(\,\pi_2\nabla_s^{2i}k\,\big)\|_{L^2(0,\mathcal L)}\leq
c(1+\|k\|_{H^m_\sharp})^{\frac{2+2\sigma}{m}}\,\,\|K\|_{H^m_\sharp}^{\frac{m-2}{m}}+\\c
(1+\|k\|_{H^{m}_\sharp}^{\frac{m-2i+\sigma}{m}})\,\|K\|_{H^m_\sharp}^{\frac{2i+\sigma}{m}}.
\end{split}\end{equation}

 Now out task is to estimate $H^r_\sharp$-norm of $K$ by
$H^r_\sharp$-norm of $k$. To this end, notice that the identity
$\nabla_s^r k=\partial_s^r K\, n$ and boundedness of $L^2$-norm of
$K=k\cdot n$ yields the estimates
\begin{equation*}\begin{split}
\|K\|_{H^r_\sharp}\leq c+\|\partial_s^r
K\|_{H^0_\sharp}=c+c\|\nabla_s^r k\|_{H^0_\sharp},
\end{split}\end{equation*}
which holds for every integer $r\geq 0$. Applying estimate
\eqref{appb54} in Corollary \ref{appb53} we arrive  at the
inequality
\begin{equation*}\begin{split}
\|K\|_{H^r_\sharp}\leq c+c\| k\|_{H^r_\sharp}.
\end{split}\end{equation*}
Substituting this inequality with $r=m$ into \eqref{appb79} leads to
the estimates
\begin{equation}\begin{split}\label{abbp80}
\|\nabla_s^{m-2-2i}
\big(\,\pi_2\nabla_s^{2i}k\,\big)\|_{H^0_\sharp}\leq
c(1+\|k\|_{H^{m}_\sharp}^{\frac{m-2i+\sigma}{m}})\,(1+\|k\|_{H^m_\sharp}^{\frac{2i+\sigma}{m}})
\\+(1+\|k\|_{H^{m}_\sharp}^{\frac{2+2\sigma}{m}})
\,(1+\|k\|_{H^m_\sharp})^{\frac{m-2}{m}}\leq
c(1+\|k\|_{H^m_\sharp})^{1+\frac{2\sigma}{m}}.
\end{split}\end{equation}
In order to complete the proof, note that  estimate \eqref{appb54}
and the interpolation inequality imply the estimate
\begin{equation*}\begin{split}
\|\nabla_s^{m-2} k\|_{H^0_\sharp}\leq c+c\| k\|_{H^{m-2}_\sharp}\leq
c+c \|k\|_{H^m_\sharp}^{1-\frac{2}{m}}.
\end{split}\end{equation*}
Combining this estimate with \eqref{abbp80} and \eqref{appb72} we
finally obtain the inequality
\begin{equation*}\begin{split}
|I_{2i}(t)|\leq c(1+\|k(t)\|_{H^m(t)})^{1+\frac{2\sigma}{m}} (1+
\|k(t)\|_{H^m_\sharp})^{1-\frac{2}{m}}\leq
c(1+\|k(t)\|_{H^m_\sharp})^{2-\frac{2(1-\sigma)}{m}},
\end{split}\end{equation*}
which along with expression \eqref{appb71} yields desired estimate
\eqref{appb70}.
\end{proof}



\subsubsection{Estimates of $\mathcal N_3$ and $\mathcal N_4$} In
order to complete the proof of Theorem \ref{theoremsobolev} it
remains to estimate the reminders $\mathcal N_3$ and $\mathcal N_4$
given by \eqref{appb24f}. The result is given by the following
lemmas.
\begin{lemma}\label{appb81}
Assume that all assumptions of Theorem \ref{theoremsobolev} are
satisfied. Let an  integer $m\geq 4$ and $\sigma\in (1/2,1)$ be
given. Then there is $c$, depending on $m$ and $\sigma$, such that
\begin{equation}\label{appb82}
|\mathcal N_3|\leq c \int_0^{t_0}
(1+\|k(t)\|_{H^{m}_\sharp}^{2-\frac{2(1-\sigma)}{m}}).
\end{equation}
\end{lemma}
\begin{proof} Recall that
\begin{equation*}
\mathcal N_3=\int_0^{t_0}\int_{\Gamma(t)}\nabla_s^{m-4}\big(\, (
k\cdot\nabla_s^2 k)\,k\big)\cdot\nabla_s^m k\, dsdt.
\end{equation*}
Hence out task is to estimate the integrand in the right hand side
of this formula. To simplify the notation we omit the symbol $t$. By
the Cauchy inequality, we have
\begin{equation}\label{appb83}
    \Big|\int_{\Gamma}\nabla_s^{m-4}((k\cdot\nabla_s^2 k)
    k)\cdot\nabla_s^m k\,ds\Big|\leq \|\nabla_s^{m-4}((k\cdot\nabla_s^2 k)
    k)\|_{H^0_\sharp}\, \|\nabla_s^m k\|_{H^0_\sharp}.
\end{equation}
Notice that $k=K\, n$ and $\nabla_s^2 k=\partial_s^2 K\, n$, which
yields
$$
(k\cdot\nabla_s^2 k) k= K^2\partial_s^2 K\,n,\quad
\nabla_s^{m-4}((k\cdot\nabla_s^2
k)k)=\partial_s^{m-4}(K^2\partial_s^2 K)\, n.
$$
From this we conclude that
\begin{equation}\label{84}
\|\nabla_s^{m-4}((k\cdot\nabla_s^2 k)
    k)\|_{H^0_\sharp}\leq \|K^2\partial_s^2 K\|_{H^{m-4}_\sharp}.
\end{equation}
The Moser inequality, and continuity of the embedding
$H^\sigma_\sharp\hookrightarrow L^\infty(0, \mathcal L)$ imply the
estimate
\begin{equation}\begin{split}\label{appb85}
\|\nabla_s^{m-4}((k\cdot\nabla_s^2 k)
    k)\|_{H^0_\sharp}\leq\\
    c \|K\|_{L^\infty(0, \mathcal L)}^2\|K\|_{H^{m-2}_\sharp}+c
    \|\partial_s^2K\|_{L^\infty(0,\mathcal L)} \|K\|_{L^\infty(0,\mathcal L)}
    \|K\|_{H^{m-4}_\sharp}\\\leq
c \|K\|_{H^\sigma_\sharp}^2\|K\|_{H^{m-2}_\sharp}+c
    \|K\|_{H^{2+\sigma}_\sharp} \|K\|_{H^\sigma_\sharp}
    \|K\|_{H^{m-4}}\,.
\end{split}\end{equation}
Since the $\|K\|_{H^0_\sharp}$ is uniformly bounded, the
interpolation inequality yields
\begin{equation*}\begin{split}
\|K\|_{H^\sigma_\sharp}\leq
c\|K\|_{H^m_\sharp}^{\frac{\sigma}{m}},\,\,\,
\|K\|_{H^{m-2}_\sharp}\leq c\|K\|_{H^m_\sharp}^{1-\frac{2}{m}},\\
\|K\|_{H^{2+\sigma}_\sharp} \leq c
\|K\|_{H^m_\sharp}^{\frac{2+\sigma}{m}},\,\,\,
 \|K\|_{H^{m-4}_\sharp}\leq c\|K\|_{H^m_\sharp}^{1-\frac{4}{m}}.
\end{split}\end{equation*}
Substituting this inequalities into \eqref{appb85} we obtain
\begin{equation}\begin{split}\label{appb86}
\|\nabla_s^{m-4}((k\cdot\nabla_s^2 k)
    k)|_{H^0_\sharp}\leq c\|K\|_{H^m_\sharp}^{1-\frac{2(1-\sigma)}{m}}\,.
\end{split}\end{equation}
Next, inequality \eqref{appb54} in Lemma \ref{appb53} leads to the
estimate
\begin{equation*}\begin{split}
\|K\|_{H^m_\sharp}\leq c+c\|\partial^m
K\|_{H^0_\sharp}=c+c\|\nabla_s^m k\|_{H^0_\sharp}\leq
c+c\|k\|_{H^m_\sharp},
\end{split}\end{equation*}
which along with \eqref{appb86} yields the estimate
\begin{equation}\begin{split}\label{appb87}
\|\nabla_s^{m-4}((k\cdot\nabla_s^2 k)
    k)|_{H^0_\sharp}\leq c(1+\|k\|_{H^m_\sharp})^{1-\frac{2(1-\sigma)}{m}}\,.
\end{split}\end{equation}
Applying again inequality \eqref{appb54} we obtain
\begin{equation*}
\|\nabla_s^m k\|_{H^0_\sharp}\leq c(+\|k\|_{H^m_\sharp}).
\end{equation*}
Substituting this inequality and inequality \eqref{appb87} we
finally arrive at the estimate
$$
\Big|\int_{\Gamma(t)}\nabla_s^{m-4}((k(t)\cdot\nabla_s^2 k(t))
    k(t))\cdot\nabla_s^m k(t)\,ds\Big|\leq
    c\,(1+\|k(t)\|_{H^m_\sharp}\,)^{2-\frac{2(1-\sigma)}{m}},
$$
which obviously yields desired estimate \eqref{appb82}.
\end{proof}
Now our task is to estimate the reminder
\begin{equation*}\label{appb90}\begin{split}
\mathcal N_4= \int_0^{t_0}\int_{\Gamma(t)}\nabla_s^{m-4}\big((k\cdot
\Upsilon)\,k)
    \cdot\nabla_s^{m}k\,dsdt
-\int_0^{t_0}\int_{\Gamma(t)} \nabla_s^{m-2}\Upsilon\cdot\nabla_s^m
k\,dsdt,
\end{split}\end{equation*}
where
$$
\Upsilon=\frac{1}{2}|k|^2\, k\,-\, k\,+\, dJ.
$$
\begin{lemma}\label{appb91}
Assume that all assumptions of Theorem \ref{theoremsobolev} are
satisfied. Let an integer $m\geq 4$ and $\sigma\in (1/2,1)$ be
given. Then there is $c$ depending on $m$ and $\sigma$, such that
\begin{equation}\label{appb92}
|\mathcal N_4|\leq c \int_0^{t_0}
(1+\|k(t)\|_{H^{m}_\sharp}^{2-\frac{2(1-\sigma)}{m}}).
\end{equation}
\end{lemma}
\begin{proof}
Introduce the functions
\begin{equation}\begin{split}\label{appb93}
M_1=-\frac{1}{2}\nabla_s^{m-2}(|k|^2k)+\frac{1}{2}\nabla_s^{m-4}(|k|^4
k),\\ M_2=-\nabla_s^{m-2}(k)+\nabla_s^{m-4}(|k|^2 k),\\ \quad
M_3=-\nabla_s^{m-2}(dJ)+\nabla_s^{m-4}((k\cdot dJ) k).
\end{split}\end{equation}
It is easily seen that
\begin{equation*}\begin{split}
\mathcal N_4=\sum_{i=1}^3\int_0^{t_0}\int_{\Gamma(t)} M_i(t)\cdot
\nabla_s^{m}k\,dsdt.
\end{split}\end{equation*}
Estimate \eqref{appb54} in Corollary \ref{appb53} implies
$$
\|\nabla_s^m k\|_{H^0_\sharp}\leq c(1+\|k\|_{H^m_\sharp}).
$$
From this and the Cauchy inequality we obtain the estimate
\begin{equation}\begin{split}\label{appb94}
|\mathcal N_4|\leq
c\sum_{i=1}^3\int_0^{t_0}\|M_i(t)\|_{H^0_\sharp(t)}\,\,
\big(\,1+\|k(t)\|_{H^m_\sharp(t)}dt\,\big)\,.
\end{split}\end{equation}
Now our task is to estimate $H^0_\sharp$-norm of $M_i$ First we
derive estimate for the quantity $M_1$. We begin with the
observation that
\begin{equation}\label{appb95}\begin{split}
    |k|^2k=K^3\,n, \quad (k\cdot (|k|^2 k))\,k= K^5\, n,\\
    \nabla_s^r(K^3 n)=\partial_s^r(K^3) \,n, \quad \nabla_s^r(K^5 n)=
    \partial_s^r(K^5) \,n,
\end{split}\end{equation} where $K=k\cdot n$ is the scalar curvature and
$r\geq 0$ is an arbitrary integer.  It follows  from this and the
Moser inequality that
\begin{equation*}
\|\nabla^{m-2}(|k|^2k)\|_{H^0_\sharp}\leq
c\|K^3\|_{H^{m-2}_\sharp}\leq
c\|K\|_{H^\sigma_\sharp}^2\|K\|_{H^{m-2}_\sharp}.
\end{equation*}
Since $\|K\|_{H^0_\sharp}$ is uniformly bounded, we may apply the
interpolation inequality to obtain
\begin{equation*}
\|\nabla^{m-2}(|k|^2k)\|_{H^0_\sharp}\leq
c\|K\|_{H^{m}_\sharp}^{\frac{2\sigma+m-2}{m}}.
\end{equation*}
Note that estimate \eqref{appb54} in Corollary \ref{appb53} and the
identity $\nabla_s^m k=\partial_s^m K\,n$ imply the inequality
\begin{equation*}\begin{split}
\|K\|_{H^m_\sharp}\leq c+c\|\partial_s^m
K\|_{H^0_\sharp}=c+c\|\nabla_s^m k\|_{H^0_\sharp}\leq
c+c\|k\|_{H^m_\sharp}.
\end{split}\end{equation*}
It follows that
\begin{equation}\label{appb95a}
\|\nabla^{m-2}(|k|^2k)\|_{H^0_\sharp}\leq
c(1+\|k\|_{H^{m}_\sharp})^{1-\frac{2(1-\sigma)}{m}}.
\end{equation}
Repeating these arguments and using identities \eqref{appb95} we
obtain
\begin{equation*}
\|\nabla^{m-4}(|k|^4k)\|_{H^0_\sharp}\leq c\,
(1+\|k\|_{H^{m}_\sharp}^{1-\frac{4(1-\sigma)}{m}}\,).
\end{equation*}
Since $\sigma\in (1/2, 1)$, we conclude from this and  expression
\eqref{appb93} for $M_1$ that
\begin{equation}\label{appb95b}
\|M_1\|_{H^0}\leq c(1+\|k\|_{H^{m}})^{1-\frac{2(1-\sigma)}{m}}.
\end{equation}
Arguing as before we obtain the estimate
\begin{equation}\label{appb96a}
\|M_2\|_{H^0_\sharp}\leq
c(1+\|k\|_{H^{m}_\sharp})^{1-\frac{4(1-\sigma)}{m}}.
\end{equation}
It remains to estimate $H^0_\sharp$-norm of $M_3$. Recall that $dJ$
is a normal field and set
$$
dJ=\psi\, n,\quad \psi=dJ\cdot n,\text{~~which yields~~} (k\cdot
dJ)k=K^2\psi\,n.
$$
Thus we get
$$
\nabla_s^{m-2} dJ= \partial_s^{m-2}\psi\, n, \quad
\nabla_s^{m-4}\big((k\cdot dJ)k)=\partial_s^{m-4}(K^2\psi)\, n,
$$
which leads to the inequality
\begin{equation*}
    \|M_3\|_{H^0_\sharp}\leq c\|\psi\|_{H^{m-2}_\sharp}+\|K^2\psi\|_{H^{m-4}}\,.
\end{equation*}
The Moser inequality and the embedding theorem imply the estimate
$$
\|M_3\|_{H^0_\sharp}\leq
c\|\psi\|_{H^{m-2}_\sharp}+c\|K\|_{H^\sigma_\sharp}^2\|\psi\|_{H^{m-4}_\sharp}+
c\|K\|_{H^\sigma_\sharp}\|\psi\|_{H^\sigma_\sharp}\|K\|_{H^{m-4}}.
$$
Since $\|\psi\|_{H^0_\sharp}=\|dJ\|_{H^0_\sharp}$ and
$\|k\|_{H^0_\sharp}$ are uniformly bounded,  we may apply the
interpolation inequality to obtain the estimate
\begin{equation}\label{appb98}
\|M_3\|_{H^0_\sharp}\leq
c\|\psi\|_{H^{m}_\sharp}^{1-\frac{2}{m}}+\|K\|_{H^m_\sharp}^{\frac{2\sigma}{m}}
\|\psi\|_{H^{m}_\sharp}^{1-\frac{4}{m}}+
\|K\|_{H^m_\sharp}^{\frac{\sigma}{m}}
\|\psi\|_{H^m_\sharp}^{\frac{\sigma}{m}}\|K\|_{H^{m}}^{1-\frac{4}{m}}.
\end{equation}
Next notice that by virtue of estimate \eqref{appb54}, we have
\begin{equation*}
\|K\|_{H^m_\sharp}\leq c(1+\|\partial_s^m K\|_{H^0_\sharp})=
c(1+\|\nabla_s^m k\|_{H^0_\sharp})\leq c(1+\|k\|_{H^m_\sharp}).
\end{equation*}
Recall that $\|dJ\|_{H^0}$ and $\|k\|_{H^0}$ are uniformly bounded.
From this and estimate \eqref{appb52} in Lemma \ref{appb50} with
$r=m$ and $\Phi=dJ$ we conclude that
\begin{equation*}
\|\psi\|_{H^m_\sharp}\leq c(1+\|\partial_s^m \psi\|_{H^0_\sharp})=
c(1+\|\nabla_s^m dJ\|_{H^0_\sharp})\leq
c(1+\|k\|_{H^m_\sharp}+\|dJ\|_{H^m_\sharp}).
\end{equation*}
By virtue of Theorem \ref{bokr60}, we have
$$
\|dJ\|_{H^{m+\beta}_\sharp}\leq c(1+\|k\|_{H^m_\sharp}),\quad
\beta\in [0,1/2),
$$
which gives
\begin{equation*}
 \|\psi\|_{H^m_\sharp}\leq c(1+\|k\|_{H^m_\sharp})\,.
\end{equation*}
Substituting the obtained estimates for $\|K\|_{H^m_\sharp}$ and
$\|\psi\|_{H^m_\sharp}$ into \eqref{appb98} and noting that
$\sigma\in(1/2,1)$ we finally obtain
\begin{equation}\label{appb99}
\|M_3\|_{H^0_\sharp}\leq (1\|k\|_{H^m_\sharp})^{1-\frac{2}{m}}\,.
\end{equation}
Combining estimates \eqref{appb95}, \eqref{appb96a} and
\eqref{appb99} we arrive at the inequality
\begin{equation*}
\sum_{i=1}^3\|M_i(t)\|_{H^0(t)}\leq
(1+\|k\|_{H^m(t)})^{1-\frac{2(1-\sigma)}{m}}\,.
\end{equation*}
Substituting this inequality into \eqref{appb94} we obtain desired
estimate \eqref{appb92}.
\end{proof}


\subsection{Proof of Theorem \ref{theoremsobolev}}
\begin{proof}
We are now in a position to complete the proof of Theorem
\ref{theoremsobolev}. We begin with the observation that the
estimates of $\mathcal N_i$ given by Lemmas \ref{appb69},
\ref{appb81}, and \ref{appb91} imply the inequality
\begin{equation*}
    \sum_{i=1}^{4}|\mathcal N_i|\leq c\int_0^{t_0}
    (1+\|k(t)\|_{H^m_\sharp}^{2-\frac{2(1-\sigma)}{m}})\, dt\,.
\end{equation*}
Since $\sigma\in (1/2,1)$ we may apply the Young inequality to
obtain
\begin{equation}\label{appb100}
 \sum_{i=1}^{4}|\mathcal N_i|\leq c_\epsilon t_0+\epsilon\int_0^{t_0}
 \|k(t)\|_{H^m_\sharp}^2\,dt,
\end{equation}
where $\epsilon\in (0,1)$ is an arbitrary number and the constant
$c_\epsilon$ depends on $\epsilon$.  Next, estimate \eqref{appb54}
in Corollary \ref{appb53} yields the inequalities
\begin{equation}\label{appb101}
\|\nabla_s^{m-2}k\|_{H^0_\sharp}\geq
(1-2\epsilon)\|k\|_{H^{m-2}_\sharp}^2-c_\epsilon, \quad
\|\nabla_s^{m}k\|_{H^0_\sharp}\geq
(1-2\epsilon)\|k\|_{H^m_\sharp}^2-c_\epsilon.
\end{equation}
Substituting these inequalities in the main integral identity
\eqref{appb24} we arrive at the estimate
\begin{equation*}\begin{split}
\frac{1-2\epsilon}{2}\|k(t_0)\|_{H^{m-2}_\sharp}^2\,
+(1-2\epsilon)\int_0^{t_0}\| k(t)\|_{H^m_\sharp}^2\,dt\leq
\\c_\epsilon (1+t_0)+ \frac{1}{2}\int_{\Gamma(0)}|\nabla_s^{m-2} k(0)|^2\,ds.
\end{split}\end{equation*}
Setting $\epsilon=1/6$ we finally obtain
\begin{equation*}\begin{split}
\|k(t_0)\|_{H^{m-2}_\sharp}^2\, +\int_0^{t_0}\|
k(t)\|_{H^m_\sharp}^2\,dt\leq\\c
(1+t_0)+\frac{3}{2}\int_{\Gamma(0)}|\nabla_s^{m-2} k(0)|^2\,ds.
\end{split}\end{equation*}
This completes the proof of  Theorem \ref{theoremsobolev}.
\end{proof}









\section{Proof of Theorem \ref{theoremsmooth}}\label{section7}
\renewcommand{\theequation}{7.\arabic{equation}}
\setcounter{equation}{0}

\begin{proof}
 Since
$k=\partial_s^2 f$, it follows from estimate \eqref{appb6} in
Theorem \ref{theoremsobolev} that
\begin{equation}\label{appb110}
    \sup\limits_{[0,T]}\, \|f(t)\|_{H^m_\sharp}\leq c(T+1)+E_m\,.
\end{equation}
Recall that the length of the curve $\Gamma(t)$ is uniformly bounded
from below and above by the constants $2/E_0$ and $E_0$. Hence the
embedding theorems holds in the spaces $H^r_\sharp(t)$, $r\geq 0$,
$t\in [0,T]$, with embedding constants independent of $t$. Since the
embedding $H^m(t)\hookrightarrow C^{m-1}(0,\mathcal L(t))$ is
continuous, it follows from this and \eqref{appb110} that $f$ and
$k$ as a function of variables $t$ and $s$ admits the estimate
\begin{equation}\label{appb111}
    \sup\limits_{[0,T]}\, \|f(t)\|_{C^{m-1}(0,\mathcal L(t))}+
    \|k(t)\|_{C^{m-3}(0,\mathcal L(t))}\leq
    c(T+1)+E_m.
\end{equation}
Notice that $s$ is an auxiliary variable and the basic independent
variable is $\theta\in \mathbb S^1$. Hence our task is to estimate
the derivatives of $f$ as a function of the variable $\theta$. To
this end, note that
\begin{equation}\label{appb112}
    \frac{\partial
    s(t,\theta)}{\partial\theta}=\sqrt{g(t,\theta)},\text{
    ~~where~~}\sqrt{g(t,\theta)}=|\partial_\theta
    f(t,\theta)|.
\end{equation}
Let us estimate the length element $\sqrt{g}$. It is easily seen
that
$$
\partial_t\sqrt{g(t)}=\frac{1}{|\partial_\theta f|}
(\partial_\theta f\cdot \partial_\theta\partial_t f) = (\partial_s
f\cdot\partial_s\partial_t f)\sqrt{g(t)}\,.
$$
Notice that
$$
\partial_s
f\cdot\partial_s\partial_t f=\tau\cdot\partial_s\partial_t f
=-\partial_s\tau\cdot \partial_t f=-k\cdot V=-\frac{1}{2}\pi_2,
$$
which yields the ordinary differential equation for $\sqrt{g}$:
\begin{equation}\label{appb114}
    \partial_t\sqrt{g}=-\frac{1}{2}\pi_2\, \sqrt{g}\text{~~~or~~~}\sqrt{g(t)}=\sqrt{g_0}\,
    e^{-\frac{1}{2}\int_0^t\pi_2dt}.
\end{equation}
It follows from estimate \eqref{appb66} in Lemma \ref{appb65} that
$$\|\pi_2\|_{H^{m-4}_\sharp}\leq
c(1+\|k\|_{H^{m-2}_\sharp})^{1+\frac{\sigma}{m-2}},
$$
which gives
\begin{equation}\label{appb115}
    \sup\limits_{[0,T]}\|\pi_2(t)\|_{C^{m-5}(0, \mathcal L(t))}\leq c.
\end{equation}
Next we have for every integer $r\geq 1$
\begin{equation*}
\partial_\theta^{r}
\pi_2=\sum_{\rho=1}^{r}\partial_s^\rho\pi_2 \sum_{\alpha}
c_{\rho\alpha}(\partial_\theta s)^{\alpha_1}\,(\partial_\theta^2
s)^{\alpha_2}\dots (\partial_\theta^r s)^{\alpha_r},
\end{equation*}
where the interior sum  is taken over all nonnegative integer
vectors $\alpha$ such that
$$
\alpha_1+\alpha_2+ \dots \alpha_r=\rho,
\quad\alpha_{1}+2\alpha_{2}+\dots+ r\alpha_{r}=r,
$$
and $c_{\rho\alpha}$ are some constants. In particular, we have
\begin{equation*}
\partial_\theta^{r}
\pi_2=\sum_{\rho=1}^{r}\partial_s^\rho\pi_2 \sum_{\alpha}
c_{\rho\alpha}( \sqrt{g})^{\alpha_1}\,(\partial_\theta
\sqrt{g})^{\alpha_2}\dots
(\partial_\theta^{r-1}\sqrt{g})^{\alpha_r}\,.
\end{equation*}
Differentiating both sides of equation \eqref{appb114} with respect
to $\theta$ we obtain
\begin{equation}\label{appb116}
    \partial_t(\partial_\theta^r\sqrt{g})=
    -\frac{1}{2}\pi_2\, (\partial_\theta^r\sqrt{g})+\mathcal R_r,
\end{equation}
where
\begin{equation*}
\mathcal
R_r=-c_\sigma\sum_{\sigma=0}^{r-1}\partial_\theta^\sigma\sqrt{g}\,
\Big(\sum_{\rho=1}^{r-\sigma}\partial_s^\rho\pi_2 \sum_{\alpha}
c_{\rho\alpha}( \sqrt{g})^{\alpha_1}\,(\partial_\theta
\sqrt{g})^{\alpha_2}\dots
(\partial_\theta^{r-\sigma-1}\sqrt{g})^{\alpha_r}\Big),
\end{equation*}
and
$$
\alpha_1+\alpha_2+ \dots \alpha_{r-\sigma}=\rho,
\quad\alpha_{1}+2\alpha_{2}+\dots+ r\alpha_{r-\sigma}=r-\sigma.
$$
Notice that $\mathcal R_r$ contains only the derivatives
$\partial_\theta^j\sqrt{g}$ of order $j\leq r-1$. Moreover by virtue
of \eqref{appb115}, we have
$$
|\partial_s^r\pi_2(t,\theta)|\leq c\text{~~for all~~} t\in
[0,T]\text{~~and~~} r\leq m-5,
$$
which yields the estimate
\begin{equation}\label{appb117}
    |R_r|\leq c \big(\,1+\sum_{i=0}^{r-1}|\partial_\theta^i \sqrt{g}|\,\big)^N,
\end{equation}
where the integer $N$ depends only on $r$. It follows from this and
\eqref{appb116} that
\begin{equation}\label{appb118}
    \sup\limits_{[0,T]\times\mathbb S^1}|\partial_\theta^r\sqrt{g}|\leq c\sup\limits_{\mathbb S^1}|\partial_\theta^r\sqrt{g_0}|+
    c \sup\limits_{[0,T]\times\mathbb S^1}\big(\,1+\sum_{i=0}^{r-1}|\partial_\theta^i
     \sqrt{g}|\,\big)^N
\end{equation}
for all $1\leq r\leq m-5$.
\\
It follows from the conditions of Theorem \ref{theoremsmooth} that
$$
\sup\limits_{\mathbb S^1}|\partial_\theta^r\sqrt{g_0}|\leq
c\text{~~for all~~} r\leq m-5\,.
$$
On the other hand, we have
$$
\sqrt{g} =\sqrt{g_0}e^{-\frac{1}{2}\int_0^t \pi_2},\text{~~which
yields~~}\sqrt{g}^{\pm 1}\leq c.
$$
From this, \eqref{appb118}, and the induction principle we conclude
that
\begin{equation}\label{appb119}
\sup\limits_{[0,T]}\|\sqrt{g(t)}\|_{C^{m-6}(\mathbb S^1)}\leq c.
\end{equation}
This inequality along with the relation
$$
\partial_\theta s(t,\theta)=\sqrt{g(t,\theta)}
$$
leads to the estimate
$$
\|s(t)\|_{C^{m-5}(\mathbb S^1)}\leq c.
$$
Combining this result with \eqref{appb11} we arrive at a priori
estimate \eqref{appb6sm} for $f$,
$$
\|f\|_{C(0,T; C^{m-5}(\mathbb S^1))}\leq c.
$$
Employing equation \eqref{gala19} and repeating the previous
arguments we obtain desired estimate \eqref{appb6sm} for $\partial_t
f$. This complete the proof of Theorem \ref{theoremsmooth}.
\end{proof}


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\section{Proof of Lemmas \ref{bokr2}, \ref{bokr3a},
and Corollary  \ref{bokr3st}}\label{lemmageom}
\renewcommand{\theequation}{8.\arabic{equation}}
\setcounter{equation}{0} {\it Proof of Lemma \ref{bokr2}.} It
suffices to  estimate $\mathcal L$ from below. The estimate from
above obviously follows from \eqref{immer11}. Since  $\Gamma $ is a
$C^1$ Jordan curve, the degree of the mapping $\tau:\Gamma\to
\mathbb S^1$ equals $1$. Hence there exists $s^*\in (0,\mathcal L)$
such that $\tau(s^*)=-\tau(0)$. We have
\begin{equation*}
2=|\tau(s^*)-\tau(0)|\leq \int_0^{s^*}|k|\, ds \leq \sqrt{\mathcal
L}\Big(\int_0^\mathcal L |k|^2\,ds\Big)^{1/2}\leq
\sqrt{2}\sqrt{\mathcal L}\sqrt{E_0},
\end{equation*}
which implies the estimate $\sqrt{\mathcal L}\geq\sqrt{2/E_0}$. The
proof of Lemma \ref{bokr2} is completed.


{\it Proof of Lemma \ref{bokr3a}.} We will consider the immersion
$f=(f_1,f_2)$ as a function of the arc-length variable $s\in
[-\mathcal L/2,\mathcal L/2]$. Obviously it can be regarded as
$\mathcal L$-periodic function on $\mathbb R$. We have $\partial_s
f_1=\tau_1$, $\partial_s f_2=\tau_2$. In the Cartesian system of
coordinates associated with $z$, we have $\tau(0)=(1,0)$. Notice
that $\tau$ is $\mathcal L$-periodic and
$$\|\partial_s\tau\|_{L^2(-\mathcal L/2,\mathcal L/2)}= \|k\|_{L^2(-\mathcal L/2,\mathcal
L/2)}\leq \sqrt{2 E_0}.
$$
It follows from this and embedding theorem that
\begin{equation}\label{bokr7}
\|\tau\|_{C^\alpha(-\mathcal L/2,\mathcal
L/2)}+\|f\|_{C^{1+\alpha}(-\mathcal L/2,\mathcal L/2)}\leq c(\alpha,
E_0)
\end{equation}
for all $\alpha\in [0,1/2)$. This means that the curve $\Gamma$
belongs to the class $C^{1+\alpha}$ and its smoothness properties
depends only on $\alpha$ and $E_0$.  In particular, there is a
positive $\kappa$, depending only on $E_0$, such that
\begin{equation*}
|\partial_s f_1(s)-1|\leq \frac{1}{12}, \quad |\partial_s
f_2(s)|\leq \frac{1}{12}\text{~~for all ~~} s\in [-3\kappa, 3\kappa]
\end{equation*}
and
\begin{equation*}
0<c(E_0)^{-1}\leq \kappa \leq c(E_0)<\infty
\end{equation*}
Therefore, the mapping $x_1=f(s)$, $s\in [-3\kappa,3\kappa]$ is
diffeomorphic. We denote its image by   $[-\alpha, \beta]$. The
mapping
$$
\eta(x_1)= f_2(f_1^{-1}(x_1)), \quad x_1\in [-\alpha, \beta]
$$
is well defined and continuously differentiable. Moreover, the
identity $\eta'=\partial_s f_2(\partial_s f_1)^{-1}$ yields the
desired estimate \eqref{bokr4} for $\eta'$. Next, we have
$$
\eta''(x_1)= (\partial_s^2 f_2(\partial_s
f_1)^{-2})(s(x_1))-2(\partial_s f_2(\partial_s f_1)^{-3}\partial_s^2
f_1)(s(x_1)).
$$
Since the absolute values of the derivatives $(\partial_s
f_1)^{\pm1}$, $\partial_s f_2$ are bounded by  $2$ on the segment
$[-3\kappa, 3\kappa]$, it follows that
$$
\int_{-\alpha}^{\beta}|\eta''|^2\, dx_1\leq c
\int_{-3\kappa}^{3\kappa}|\partial_s^2 f|^2 ds\leq c E_0.
$$
This completes the proof of Lemma \ref{bokr3a}.

{\it Proof of Corollary \ref{bokr3st}.  }
 By virtue of Corollary \ref{bokr3s}, we have for every $t\in[0,T]$,
\begin{equation*}
\nu(t)=\inf\limits_{z\in \Gamma(t)} \text{~dist~}
(\Gamma(t)\setminus \Gamma_{3\kappa}(t), \,
\Gamma_{2\kappa}(t))>0\,.
\end{equation*}
Here  $\Gamma_{3\kappa}(t)$ and  $\Gamma_{2\kappa}(t))$ are the arcs
centered at $z$ and defined by Lemma \ref{bokr3a} with $\Gamma$
replaced by $\Gamma(t)$.

It is necessary to prove that $\inf_t \nu(t) >0$. Suppose, in
contrary to the our claim, that there are sequences $t_n$ and
$z_n\in \Gamma(t_n)$  with the properties
$$
\text{~dist~} (\Gamma(t_n)\setminus \Gamma^n_{3\kappa}(t_n), \,
\Gamma^n_{2\kappa}(t_n))\to 0\text{~~as~~} n\to\infty.
$$
Here $\Gamma^n_{3\kappa}(t_n)$ and $\Gamma^n_{2\kappa}(t_n)$ are the
arcs centered at $z_n$ and defined by Lemma \ref{bokr3a} with
$\Gamma$ replaced by $\Gamma(t_n)$.

After passing to a subsequence we may assume that $t_n\to
t_\infty\in [0,T]$ and $z_n\to z^*\in\mathbb R^2$ as $n\to \infty$.
It follows that there are sequences $z_n'\in \Gamma(t_n)\setminus
\Gamma_{3\kappa}(t_n)$ and $z_n''\in \Gamma_{2\kappa}(t_n)$ such
that $|z_n'-z_n''|\to 0$ as $n\to\infty$. Choose the arc-length
coordinates on $\Gamma(t_n)$ such that the corresponding arc-length
coordinates equal zero at $z^*$. It follows that
$$
z_n'={f}(t_n, s_n'), \quad z_n''={f}(t_n, s_n''),
$$
where
 $$
 s_n'\in (-\mathcal L(t_n)/2, \mathcal L(t_n)/2)\setminus(-3\kappa,
 3\kappa), \quad s_n''\in (-2\kappa, 2\kappa).
 $$
Passing to subsequences we may assume that
$$
 s_n'\to s'_\infty\in (-\mathcal L(t_\infty)/2, \mathcal L(t_\infty)/2)\setminus(-3\kappa,
 3\kappa), \quad s_n''\to s"_\infty\in (-2\kappa, 2\kappa).
 $$
It follows from condition {\bf H.3} that
$$
z_n'=\overline{f}(t_n, s_n')\to \overline{ f}(t_\infty,
s_\infty')\text{~~and~~} z_n''=\overline{f}(t_n, s_n'')\to
\overline{ f}(t_\infty, s_\infty'').
$$
Hence \begin{equation}\label{bokr3rt}
 { f}(t_\infty, s_\infty')=
{ f}(t_\infty, s_\infty'')\text{~~and~~} s_\infty'\neq s_\infty''.
\end{equation}
On the other hand, Condition \textbf{H.2} implies that the curve
$\Gamma(t_\infty)$ has no self-intersections. This contradicts to
relations \eqref{bokr3rt}.



\section{Sobolev spaces}\label{sobolspace}
\renewcommand{\theequation}{9.\arabic{equation}}
\setcounter{equation}{0}
\subsection{Anisotropic spaces}\label{anisotropa}







 In the proof of basic results in  Section \ref{model}, the analysis
 of function classes with specific smoothness properties with
 respect to specific space variables is required.
In this subsection we collect the basis facts about such spaces. The
results will be used in Sections \ref{indbasis} and \ref{lemma28}.


Let
$$
Q_m= (-\kappa_m,\kappa_m)\times (-\rho_m, \rho_m), \quad m\in [1,r]
$$
be a rectangle defined by \eqref{grad2}. Fix an arbitrary  smooth
function $\phi:Q_m\to\mathbb R$ compactly supported in the interval
$y_1\in (-\kappa_m,\kappa_m)$ for every $y_2\in (-\rho_m,\rho_m)$.
Assume that $\phi$ is extended by $0$ to the strip $\mathbb R\times
(-\rho_m,\rho_m)$. Introduce the norm
\begin{equation}\label{anisotropic1}\begin{split}
    \|\phi\|_{X^\infty}=\sup\limits_{y_1\in\mathbb
    R}\,\,\|\phi(y_1,\cdot)\|_{L^2(-\rho_m,\rho_m)},\\
\|\phi\|_{Y^{j}}^2=\|\phi\|_{L^2(Q_m)}^2+
\|\partial_1^j\phi\|_{L^2(Q_m)}^2. \end{split}\end{equation}
Since
$\phi(\cdot, y_2)$ is smooth and compactly supported in $\mathbb R$,
we can rewrite this definition in terms of the Fourier transform.
The Fourier transform $\hat{\phi}(\xi,y_2)$, $(\xi, y_2)\in \mathbb
R\times(-\rho_m, \rho_m)$, is defined by the equalities
\begin{equation*}
    \hat{\phi}(\xi,y_2)=\frac{1}{\sqrt{2\pi}}\int_{\mathbb R}
    e^{-i\xi y_1}\phi(y_1,y_2)\, dy_1, \quad \phi(y_1,y_2)=\frac{1}{\sqrt{2\pi}}\int_{\mathbb R}
    e^{i\xi y_1}\hat{\phi}(\xi,y_2)\, d\xi.
\end{equation*}
With this notation definition \eqref{anisotropic1} for $Y^j$-norm
can be written in the equivalent form
\begin{equation}\label{anisotropic2}\begin{split}
\|\phi\|_{Y^{j}}^2=\int_{-\rho_m}^{\rho_m}\Big( \int_{\mathbb
R}(1+\xi^2)^{j}\,|\hat{\phi}(\xi,y_2)|^2\,d\xi\Big)\, dy_2.
\end{split}\end{equation}
Using \eqref{anisotropic2} we can define $Y^\beta$-norm for any
$\beta\in \mathbb R$:
\begin{equation}\label{anisotropic3}\begin{split}
\|\phi\|_{Y^{\beta}}^2=\int_{-\rho_m}^{\rho_m}\Big( \int_{\mathbb
R}(1+\xi^{2})^\beta\,\,|\hat{\phi}(\xi,y_2)|^2\,d\xi\Big)\, dy_2.
\end{split}\end{equation}
Notice two elementary inequalities for the introduced norms.
\begin{lemma}\label{anisotropic10}
 Let
$\phi$ be a smooth function compactly supported in $Q_m$. Assume
that $\phi$ is extended by $0$ to $\mathbb R\times
(-\rho_m,\rho_m)$. Then
\begin{equation}\label{anisotropic4}
    \|\phi\|_{X^\infty}\leq
    c(\sigma)\,\,\|\phi\|_{Y^\sigma}\text{~~for every~~}\sigma>1/2,
\end{equation}
\begin{equation}\label{anisotropic5}
    \|\phi\|_{Y^\gamma}\leq
    c(\sigma)\,\,\|\phi\|_{L^2(Q_m)}^{1-\frac{\gamma}{\sigma}}\,\,
    \|\phi\|_{Y^\sigma}^{\frac{\gamma}{\sigma}}\text{~~for every~~}\sigma> 0
    \text{~~and~~}0<\gamma<\sigma.
\end{equation}
\end{lemma}
\begin{proof}


In order to prove the first estimate \eqref{anisotropic4} in Lemma
\ref{anisotropic10}, note that
$$
\sup\limits_{y_1\in\mathbb R} |\phi(y_1,y_2)|\leq
\frac{1}{\sqrt{2\pi}}\,\int_R |\hat{\phi} (\xi,y_2)|\, d\xi.
$$
Thus we get
\begin{equation}\label{anisotropic6}\begin{split}
\|\phi\|_{X^\infty}^2=\sup\limits_{y_1\in\mathbb
R}\int_{-\rho_m}^{\rho_m} |\phi(y_1,y_2)|^2\, dy_2\leq
\int_{-\rho_m}^{\rho_m}\Big\{\sup\limits_{y_1\in\mathbb
R}|\phi(y_1,y_2)|\Big\}^2\, dy_2\leq \\
c\int_{-\rho_m}^{\rho_m}\Big\{\int_{\mathbb
R}|\hat{\phi}(\xi,y_2)|\, d\xi\Big\}^2\, dy_2.
\end{split}\end{equation}
Since $\sigma>1/2$, the Cauchy inequality implies the estimate
\begin{equation*}\begin{split}
\Big\{\int_{\mathbb R}|\hat{\phi}(\xi,y_2)|\, d\xi\Big\}^2\leq
\int_{\mathbb R}(1+\xi^2)^{-\sigma}\, d\xi \,\,\int_{\mathbb R}
(1+\xi^2)^\sigma\,\,\|\hat{\phi}(\xi,y_2)|^2\, d\xi\\
\leq c \,\int_{\mathbb R}
(1+\xi^2)^\sigma\,\,|\hat{\phi}(\xi,y_2)|^2\, d\xi.
\end{split}\end{equation*} Substituting this estimate into
\eqref{anisotropic6} we finally arrive at the inequality
\begin{equation*}
\|\phi\|_{X^\infty}^2\leq \int_{-\rho_m}^{\rho_m}\int_{\mathbb R}
(1+\xi^2)^\sigma\,\,|\hat{\phi}(\xi,y_2)|^2\, d\xi\, dy_2,
\end{equation*}
which along with \eqref{anisotropic3} yields the desired estimate
\eqref{anisotropic4}.

It remains to prove the interpolation inequality
\eqref{anisotropic5}. We
 begin with the observation that
$$
(1+\xi^2)^\gamma |\hat{\phi}(\xi,y_2)|^2=
(|\hat{\phi}(\xi,y_2)|^2)^{1-\vartheta}\,\big(\,(1+\xi^2)^\gamma
|\hat{\phi}(\xi,y_2)|^2)^\vartheta\,
\big),\,\,\,\,\,\vartheta=\gamma/\sigma.
$$
From this and the H\"{o}lder inequality we conclude that
\begin{equation*}\begin{split}\label{anisotropic9}
\int_{\mathbb R}(1+\xi^2)^\gamma |\hat{\phi}(\xi,y_2)|^2\, d\xi\leq
\Big(\,\int_{\mathbb R}|\hat{\phi}(\xi,y_2)|^2\,
d\xi\,\Big)^{1-\vartheta}\,\Big(\,\int_{\mathbb
R}(1+\xi^2)^\sigma|\hat{\phi}(\xi,y_2)|^2\,
d\xi\,\Big)^{\vartheta}\,.
\end{split}\end{equation*}
Combining this inequality with \eqref{anisotropic2} and applying the
H\"{o}lder inequality we finally arrive at the  estimate
\begin{equation*}\begin{split}
\|\phi\|_{Y^\gamma}^2=\int_{-\rho_m}^{\rho_m}\Big\{\int_{\mathbb
R}(1+\xi^2)^\gamma|\hat{\phi}(\xi,y_2)|^2\, d\xi\Big\}^2\,dy^2\leq\\
\int_{-\rho_m}^{\rho_m}\Big\{\Big(\,\int_{\mathbb
R}|\hat{\phi}(\xi,y_2)|^2\,
d\xi\,\Big)^{1-\vartheta}\,\Big(\,\int_{\mathbb
R}(1+\xi^2)^\sigma|\hat{\phi}(\xi,y_2)|^2\,
d\xi\,\Big)^{\vartheta}\Big\}\, dy_2\leq\\
\Big(\int_{-\rho_m}^{\rho_m}\int_{\mathbb
R}|\hat{\phi}(\xi,y_2)|^2\, d\xi dy_2\Big)^{1-\vartheta}\,
\Big(\int_{-\rho_m}^{\rho_m}\int_{\mathbb
R}(1+\xi^2)^\sigma|\hat{\phi}(\xi,y_2)|^2\,
d\xi dy_2\Big)^{\vartheta}=\\
\|\phi\|_{Y^0}^{2(1-\vartheta)}\,
\|\phi\|_{Y^\sigma}^{2\vartheta}\,.
\end{split}\end{equation*}
Recalling that $\vartheta=\gamma/\sigma$ and
$$
\|\phi\|_{Y^0}^2=\int_{-\rho_m}^{\rho_m}\int_{\mathbb
R}|\hat{\phi}(\xi,y_2)|^2\, d\xi
dy_2=\int_{-\rho_m}^{\rho_m}\int_{\mathbb R}|\phi(y)|^2\,
dy_1dy_2=\|\phi\|_{L^2(Q_m)}^2
$$
we obtain the desired interpolation inequality \eqref{anisotropic5}.
\end{proof}
\subsection{Proof of Lemma \ref{unit8}}\label{extension} Without loss
of generality we may assume that $\mathcal L=2\pi$.  For the sake of
simplicity, introduce the temporary notation:
\begin{equation}\label{extension1}
    \overline{F}=u, \quad F_\sharp=v=\sum_{k=-\infty}^{\infty}
    u(s+2k\pi).
\end{equation}
It is necessary to prove that
\begin{equation}\label{extension2}
    \|v\|_{H^{m+1/2}_\sharp}\leq \|u\|_{H^{m+1/2}(\mathbb R)}.
\end{equation}
{\it Step 1. New norm in $H^{m+1/2}(\mathbb R)$. } It is convenient
to introduce the equivalent norm on the space $H^{m+1/2}(\mathbb
R)$. Recall that
\begin{equation}\label{extension3}
\|u\|_{H^{m+1/2}(\mathbb R)}^2=\int_{\mathbb
R}(1+|\xi|^2)^{m+1/2}\,|\hat{u}(\xi)|^2\, d\xi.
\end{equation}
Introduce the pseudodifferential operators $\mathbf S$ and $\mathbf
T$ defined in terms of the Fourier transform by the equalities
\begin{equation}\label{extension4}
\widehat{\mathbf S\,u}(\xi)=\frac{1}{(1+|\xi|^2)^{1/4}}\,\,\hat
{u}(\xi), \quad \widehat{\mathbf T\,u}(\xi)= \mathcal
T(\xi)\,\hat{u}(\xi),
\end{equation}
where
\begin{equation}\label{extension4a}
\mathcal T(\xi)=\frac{(-i\xi)^{m+1}}{(1+|\xi|^2)^{1/4}}\,\,\hat
{u}(\xi).
\end{equation}
It is clear that
\begin{equation}\label{extension5}
\mathbf T=(\partial_s)^{m+1}\,\, \mathbf S.
\end{equation}
Introduce the Hilbert norm defined by
\begin{equation}\label{extension6}
    |u |_{H^{m+1/2}(\mathbb R)}^2=\|u\|_{L^2(\mathbb R)}^2+\|\mathbf T\,u\|_{L^2(\mathbb
    R)}^2,
\end{equation}
or equivalently
\begin{equation}\label{extension7}
    |u |_{H^{m+1/2}(\mathbb R)}^2=\int_{\mathbb R}(1+|\mathcal
    T(\xi)|^2)\,|\hat{u}(\xi)|^2\, d\xi.
\end{equation}
Since
$$
c^{-1}(1+|\xi|^2)^{m+1/2}\leq 1+|\mathcal T(\xi)|^2\leq
c(1+|\xi|^2)^{m+1/2},
$$
the norms $\|\cdot\|_{H^{m+1/2}(\mathbb R)}$ and
$|\cdot|_{H^{m+1/2}(\mathbb R)}$ are equivalent, i.e.,
\begin{equation}\label{extension8}
c^{-1}\|u\|_{H^{m+1/2}(\mathbb R)}\leq |u|_{H^{m+1/2}(\mathbb R)}
\leq c\|u\|_{H^{m+1/2}(\mathbb R)}.
\end{equation}
Finally introduce the function
\begin{equation}\label{extension9}
    \Phi(\xi)=\mathcal T(\xi)\, \hat{u}(\xi), \quad \xi\in \mathbb
    R.
\end{equation}
It is clear that
\begin{equation}\label{extension10}
|u |_{H^{m+1/2}(\mathbb R)}^2=\|\hat{u} \|_{L^2(\mathbb R)}^2+\|
\Phi\|_{L^2(\mathbb R)}^2.
\end{equation}






{\it Step 2. New norm in $H^{m+1/2}_\sharp$. } It is convenient to
introduce the equivalent norm on the space $H^{m+1/2}_\sharp$.
Recall that
\begin{equation}\label{extension12}
\|v\|_{H^{m+1/2}_\sharp}^2=\sum_{k=-\infty}^{\infty}(1+|k|^2)^{m+1/2}\,|v_k|^2,
\end{equation}
where the Fourier coefficients are defined by
$$
v_k=\frac{1}{\sqrt{2\pi}}\int_{-\pi}^\pi e^{-iks}\, v(s)\, ds.
$$
Introduce the  operators $\mathbf S_\sharp$ and $\mathbf T_\sharp$
defined in the Fourier basis  by the equalities
\begin{equation}\label{extension13}
(\mathbf S_\sharp\,v)_k=\frac{1}{(1+|k|^2)^{1/4}}\,v_k, \quad
(\mathbf T_\sharp\,v)_k=\mathcal T(k)\,v_k,
\end{equation}
where $\mathcal T$ is defined by \eqref{extension4a}. It is clear
that
\begin{equation}\label{extension14}
\mathbf T_\sharp=(\partial_s)^{m+1}\,\, \mathbf S_\sharp.
\end{equation}
Introduce the Hilbert norm defined by
\begin{equation}\label{extension15}
    |v |_{H^{m+1/2}_\sharp}^2=\sum_k |v_k|^2+\sum_k|\mathcal T(k)|^2
    \,|v_k|^2.
\end{equation}
Arguing as before we conclude that  the norms
$\|\cdot\|_{H^{m+1/2}\sharp}$ and $|\cdot|_{H^{m+1/2}_\sharp}$ are
equivalent, i.e.,
\begin{equation}\label{extension16}
c^{-1}\|v\|_{H^{m+1/2}_\sharp}\leq |v|_{H^{m+1/2}_\sharp} \leq
c\|v\|_{H^{m+1/2}_\sharp}.
\end{equation}
Finally introduce the sequence
\begin{equation}\label{extension17}
    \Phi_\sharp(k)=\mathcal T(k)\, v_k, \quad -\infty<k<\infty.
\end{equation}
It is clear that
\begin{equation}\label{extension18}
|v |_{H^{m+1/2}_\sharp}^2=\sum_k |v_k|^2+\sum_k
    \,|\Phi_\sharp (k)|^2.
\end{equation}
Hence we can rewrite inequality \eqref{extension2} in the form
\begin{equation*}
    |v |_{H^{m+1/2}_\sharp}\leq c |u |_{H^{m+1/2}(\mathbb R)}.
\end{equation*}
From this and representations \eqref{extension10},
\eqref{extension18} for the norms in the spaces $H^{m+1/2}_\sharp$
and $H^{m+1/2}(\mathbb R)$ we conclude that this equality is
equivalent to the following
\begin{equation}\label{extension19}
\sum_k |v_k|^2+\sum_k
    \,|\Phi_\sharp (k)|^2\leq c(\|\hat{u} \|_{L^2(\mathbb R)}^2+\|
\Phi\|_{L^2(\mathbb R)}^2).
\end{equation}

{\it Step 3. Relation between $\Phi$ and $\Phi_\sharp$.} Let  a
compactly supported function $u$ and a periodic function $v$ are
connected by relation \eqref{extension1},
\begin{equation*}
v=\sum_{k=-\infty}^{\infty}
    u(s+2k\pi).
\end{equation*}
 Furthermore assume that $\Phi$ and $\Phi_\sharp$ are defined by
\eqref{extension9} and \eqref{extension17}. Then we have the
identity
\begin{equation}\label{extension20}
    \Phi_\sharp(k)=\Phi(k), \quad -\infty<k<\infty.
\end{equation}
Indeed, we have
\begin{equation*}\begin{split}
\Phi_\sharp(k)=\mathcal T(k) \, v_k=\mathcal T(k)
\frac{1}{\sqrt{2\pi}}\int_0^{2\pi}e^{-iks}v(s)\, ds=\\
\sum_{j=-\infty}^\infty\frac{\mathcal
T(k)}{\sqrt{2\pi}}\int_{0}^{2\pi}e^{-iks} u(s+2\pi j)\, ds \mathcal
=\sum_{j=-\infty}^\infty\frac{\mathcal T(k)}{\sqrt{2\pi}}\int_{2\pi
j}^{2\pi(j+1)}e^{-iks}u(s)\, ds\\=\mathcal
T(k)\frac{1}{\sqrt{2\pi}}\int_{\mathbb R}e^{-iks}u(s)\, ds=\Phi(k).
\end{split}\end{equation*}
Similarly we have
\begin{equation}\label{extension21}
    v_k=\hat{u}(k), \quad -\infty<k<\infty.
\end{equation}
Substituting  \eqref{extension20} and \eqref{extension21} into
\eqref{extension19} we conclude that it suffices to prove the
inequality
\begin{equation}\label{extension22}
\sum_k (|\hat{u}(k)|^2+
    \,|\Phi (k)|^2)\leq c \int_{\mathbb R}
    (|\hat{u}(\xi) |^2+|
\Phi(\xi)|^2)\,d\xi.
\end{equation}
{\it Step 4.} The proof of inequality \eqref{extension22} is based
on the following lemma
\begin{lemma}\label{extension23}
Under the above assumptions we have
\begin{equation}\label{extension24}
    \|\Phi\|_{H^{1}(\mathbb R)}\leq c\|\hat{u}\|_{L^2(\mathbb R}+c\|\Phi\|_{L^2(\mathbb R)}, \quad
    \|\hat{u}\|_{H^{1}(\mathbb R)}\leq c \|\hat{u}\|_{L^2(\mathbb
    R)}.
\end{equation}
\end{lemma}
\begin{proof}Recall that
$$
\mathbf T u=\partial_s^{m+1}\mathbf Su, \quad \widehat{\mathbf S u}=
(1+|\xi|^2)^{-1/4}\,\hat{u}.
$$
The operator $\mathbf S$ is the Bessel potential of order $1/2$. It
admits the integral representation
\begin{equation}\label{extension25}
\mathbf S\, u(s)= \int_{\mathbb R} \mathbf b(s-t)\, u(t)\, dt,
\end{equation}
where the Bessel kernel $\mathbf b$ has the following properties,
see \cite{ADAMS}, Ch. 1. On the interval $(-\kappa,\kappa)$ it has
the representation
\begin{equation}\label{extension26}
 \mathbf b(s)= \frac{c(\kappa)}{\sqrt{|s|}}+o(s), \quad o\in
 C^\infty[-\kappa,\kappa],
\end{equation}
Outside of this interval the kernel $\mathbf b(s)$ admits the
estimate
\begin{equation}\label{extension27}
 |\partial_s^r\mathbf b(s)|\leq c(r)e^{-\beta |s|}\text{~~for all~~}
 |s|\geq \kappa,\,\,\,r\geq 0.
\end{equation}
Here $\beta$ is some positive exponent. Choose an arbitrary function
$\zeta\in C^\infty_0(\mathbb R)$ such that
$$
\zeta(s)=1\text{~~for~~}|s|\leq 2\kappa, \quad
\zeta(s)=0\text{~~for~~}|s|\geq 3\kappa.
$$
We have
$$
\mathbf Tu=\zeta\mathbf Tu+(1-\zeta)\mathbf Tu
$$
Obviously we have
\begin{equation}\label{extension30}
    \int_{\mathbb R} |s\,\zeta\,\mathbf Tu|^2\,ds\leq
    c\kappa^2\int_{\mathbb R}|\mathbf Tu|^2\, ds.
\end{equation}
Recall that $u$ is compactly supported in the interval
$(-\kappa,\kappa)$. From this and estimate \eqref{extension27} we
conclude that the inequalities
\begin{equation*}\begin{split}
|\mathbf Tu(s)|=\Big|\int_{\mathbb R}\partial_s^{m+1}\mathbf b(s-t)
u(t)\,dt\Big|\leq\\c\int_{\mathbb R}e^{-\beta|s-t|} u(t)\,dt\leq c
e^{-\beta|s|}\int_{\mathbb R} |u| \,dt\leq c
e^{-\beta|s|}\,\|u\|_{L^2(\mathbb R)}
\end{split}\end{equation*}
hold for every $s$ with $|s|\geq 2\kappa$.

Since $(1-\zeta)\,\mathbf Tu$ equals zero in the segment
$[-2\kappa,2\kappa]$, it follows from this that
\begin{equation*}\begin{split}
\int_{\mathbb R}|s(1-\zeta)\,\mathbf Tu(s)|^2\, ds\leq c
\int_{\mathbb R}
s^2\,e^{-2\beta|s|}\,ds\,\,\|u\|_{L^2(\mathbb R)}^2\\
\leq c \|u\|_{L^2(\mathbb R)}^2.
\end{split}\end{equation*}
Combining this estimate with \eqref{extension30} we obtain
\begin{equation}\label{extension31}
\int_{\mathbb R}|s\mathbf T u|^2\, ds \leq
c\big(\,\|u\|_{L^2(\mathbb R)}^2+\| \mathbf Tu\|_{L^2(\mathbb
R)}^2\,\big).
\end{equation}
Next notice that
\begin{equation*}
\widehat{s\mathbf Tu}=\frac{1}{i}\partial_\xi \widehat{\mathbf Tu}=
\frac{1}{i}\partial_\xi \Phi, \quad \widehat{\mathbf Tu}=\Phi.
\end{equation*}
From this, \eqref{extension31}, and the Plancherel theorem we obtain
\begin{equation*}
\int_{\mathbb R}|\partial_\xi \Phi|^2\, d\xi \leq c\big(\,\|\hat{
u}\|_{L^2(\mathbb R)}^2+\| \Phi\|_{L^2(\mathbb R)}^2\,\big),
\end{equation*}
which gives the desired estimate \eqref{extension24} for $\Phi$.
Repeating these arguments with essential simplifications we finally
obtain estimate \eqref{extension24} for $\hat{u}$, and the lemma
follows.\end{proof}

We are now in a position to  complete the proof of Lemma
\ref{unit8}. To this end, it suffices to prove inequality
\eqref{extension22}. Notice that
$$
|\Phi(k)|^2\leq 2|\Phi(\xi)|^2+2|\Phi(\xi)-\Phi(k)|^2.
$$
Next we have
$$
|\Phi(\xi)-\Phi(k)|\leq \Big(\int_k^{k+1} |\partial_\xi\Phi|^2\,
d\xi\Big)^{1/2}\text{~~for~~} \xi\in [k,k+1].
$$
It follows that
$$
|\Phi(k)|^2\leq 2|\Phi(\xi)|^2+2\int_k^{k+1} |\partial_\xi\Phi|^2\,
d\xi\text{~~for~~} \xi\in [k,k+1].
$$
Integrating both sides of these inequality over the segment
$[k,k+1]$ we arrive at the inequality
$$
|\Phi(k)|^\leq 2\int_k^{k+1}(|\Phi(\xi)|^2+ |\partial_\xi\Phi|^2\,
d\xi.
$$
Summation this inequality with respect to $k$ gives
\begin{equation*}
    \sum_{-\infty}^\infty |\Phi(k)|^2\leq 2\int_{\mathbb R}(|\Phi(\xi)|^2+ |\partial_\xi\Phi|^2)\,
d\xi=\|\Phi\|_{H^1(\mathbb R)}^2.
\end{equation*}
From this and Lemma \ref{extension23} we conclude that
\begin{equation}\label{extension33}
    \sum_{-\infty}^\infty |\Phi(k)|^2\leq c\,(\|\hat{u}\|_{L^2(\mathbb R)}^2
    +\|\Phi\|_{L^2(\mathbb R)}^2).
\end{equation}
Repeating these arguments gives
\begin{equation}\label{extension34}
    \sum_{-\infty}^\infty |\hat{u}(k)|^2\leq c\,\|\hat{u}\|_{L^2(\mathbb R)}^2.
\end{equation}
Combining \eqref{extension33} and \eqref{extension34} we obtain
desired inequality \eqref{extension22}. This completes the proof of
Lemma \ref{unit8}.

\section{Proof of Lemma \ref{tran15}}\label{indbasis}
\renewcommand{\theequation}{10.\arabic{equation}}
\setcounter{equation}{0}
 We begin with the observation that the function $u$ satisfies the
divergent elliptic equation
$$
\text{div~}(aN\nabla u)=0\text{~~in~~} Q_0
$$
 with the  matrix $aN$ bounded
from below and above by the constants, depending on  $C_N$ and
$a_0$. It follows from this, inequality
\begin{equation*}
    \|u\|_{L^2(Q_0)}+\|\nabla u\|_{L^2(Q_0)}\leq C_u<\infty
\end{equation*}
 and di-Giorgi-Nash-Moser
estimate for the H\"{o}lder norm of solutions to the divergent
elliptic equations  that the estimate
\begin{equation*}
    \|u\|_{C^\alpha(Q')}\leq c(Q')\text{~~for some ~~}\alpha\in
    [0,1),
\end{equation*}
holds for every compact set $Q'\Subset Q_0$. We thus get the
estimate
\begin{equation}\label{tran20}
    \|v_1\|_{C(Q_1)}\leq c\,\|u\|_{C(Q_1)}\leq c.
\end{equation}
For $m=1$, equation \eqref{tran14a} reads
\begin{equation}\label{tran21}\begin{split}
    \text{div~}\big(a N\nabla\partial_1 v_1)=
    -\text{~div~}(a\partial_1 N\nabla v_1)+\\\text{~div~}\big(\partial_1
    u\,aN\nabla\varphi_1+u\,a\partial_1(N\nabla\varphi_1)\big)+\partial_1(a\nabla\varphi_1\cdot
    N\nabla u) \text{~~in~~} Q_1.
\end{split}\end{equation}
Now we estimate every term in the right hand side of this equation.
Since $a$ is bounded and $N$ is independent of $y_2$, we have
\begin{equation*}\begin{split}
\|a\partial_1 N\nabla v_1\|_{L^2(Q_1)}\leq c\,\|\partial_1 N\nabla
v_1\|_{L^2(Q_1)}\leq \\
c\|\partial_1 N\|_{L^2(-\kappa_1,\kappa_1)}\sup\limits_{y_1\in
(-\kappa_1,\kappa_1)} \| \nabla v_1\|_{L^2(-\rho_1,\rho_1)}=\\
c\,\|\partial_1 N\|_{L^2(-\kappa_1,\kappa_1)}\, \|\nabla
v_1\|_{X^\infty}\leq  c\, \|\nabla v_1\|_{X^\infty}.
\end{split}\end{equation*}
Here we use estimate \eqref{tran5} for $\partial_1 N$. From this and
inequalities \eqref{tran4}-\eqref{tran5} with $\phi=\nabla v_1$ we
conclude that for every fixed $\sigma\in (0,1)$,
\begin{equation}\label{tran22}\begin{split}
\|a\partial_1 N\nabla v_1\|_{L^2(Q_1)}\leq c\,  \|\nabla
v_1\|_{X^\infty}\leq c\,\|\nabla v_1\|_{Y^\sigma}\\\
\leq c\,\|\nabla v_1\|_{L^2(Q_1)}^{1-\sigma}\|\nabla
v_1\|_{Y^1}^\sigma\leq c\, \|\nabla v_1\|_{Y^1}^\sigma.
\end{split}\end{equation}
Let us estimate the second term in the right hand side of
\eqref{tran21}. Recall that $L^2$-norm of $\nabla u$ and
$\partial_1N$ are uniformly bounded by the constants  $C_N$ and
$C_u$. In its turn,  $L^\infty$ norms of $u$ and $N$ are bounded by
$C_N$ and the constant in inequality \eqref{tran20}. It follows from
this that
\begin{equation}\label{tran23}\begin{split}
\|a\partial_1 u\, N\nabla\varphi_1+
 u\,a\,\partial_1(N \nabla\varphi_1)\|_{L^2(Q_1)}\leq\\ c\,\|\nabla
 u\|_{L^2(Q_1)}+c\,\|\partial_1(N\nabla\varphi_1)\|_{L^2(Q_1)}\leq c.
\end{split}\end{equation}
It remains to estimate the third term in the right hand side of
\eqref{tran21}. We have
\begin{equation}\label{tran24}\begin{split}
\|a\nabla\varphi_1\cdot N\nabla u\|_{L^2(Q_1)}\leq c\,\|\nabla
 u\|_{L^2(Q_1)}\leq c.
\end{split}\end{equation}
  Multiplying both sides of
\eqref{tran21} by $\partial_1 v_1$ and integrating the result by
parts we arrive at the equality
\begin{equation*}\begin{split}
\int_{Q_1}aN\nabla\partial_1 v_1\cdot \nabla\partial_1 v_1dy
=-\int_{Q_1} a\,\partial_1N\nabla v_1\cdot \nabla \partial_1 v_1\, dy+\\
\int_{Q_1}(\partial_1u\,\,a\,N\nabla\varphi_1+
u\,a\,\partial_1(N\nabla\varphi_1))\cdot\nabla \partial_1v_1\, dy+
\int_{Q_1}(a\nabla\varphi_1\cdot N\nabla u)\,\partial_1^2v_1\, dy.
\end{split}\end{equation*}
Applying the Cauchy inequality in the right hand side and employing
estimates \eqref{tran22}-\eqref{tran24} we obtain
\begin{equation}\begin{split}\label{tran25}
\int_{Q_1}aN\nabla\partial_1 v_1\cdot \nabla\partial_1 v_1dy \leq
c\,(\|\nabla v_1\|_{Y^1}^{\sigma}+1)\,\,\|\nabla\partial_1
v_1\|_{L^2(Q_1)}.
\end{split}\end{equation}
Recall that the matrix $aN$ is bounded from below by the constant
depending only on $C_N$ and $a_0$, which gives
$$
\int_{Q_1}aN\nabla\partial_1 v_1\cdot \nabla\partial_1 v_1dy\geq
c^{-1}\|\nabla \partial_1 v_1\|_{L^2(Q_1)}^2.
$$
Obviously we have
\begin{multline*}
\|\nabla v_1\|_{Y^1}^\sigma\leq c(\|\nabla_1 v_1\|_{L^2(Q_1)}+
\|\partial_1\nabla_1 v_1\|_{L^2(Q_1)})^\sigma\\\leq  c(1+
\|\partial_1\nabla_1 v_1\|_{L^2(Q_1)})^\sigma\leq c +c
\|\partial_1\nabla v_1\|_{L^2(Q_1)}^\sigma.
\end{multline*}
Substituting the obtained estimates into \eqref{tran25} we arrive at
the inequality
\begin{equation*}\begin{split}
\|\nabla\partial_1 v_1\|_{L^2(Q_1)} \leq c(\|\nabla\partial_1
v_1\|_{L^2(Q_1}^\sigma+c,\quad \sigma\in (0,1),
\end{split}\end{equation*}
which obviously yields the desired estimate \eqref{tran16}
\begin{equation*}
    \|\partial_1 \nabla v_1\|_{L^2(Q_1)}\leq c.
\end{equation*}





\section{Proof of Lemma \ref{tran28}}\label{lemma28}
\renewcommand{\theequation}{11.\arabic{equation}}
\setcounter{equation}{0}



\subsection{Proof of estimate
\eqref{tran29} for $\mathcal A_m$.}Introduce the temporary notation
$$
H=\partial_1 N, \quad w=\partial_1 \nabla v_m.
$$
Since $\varphi_1=1$ on the support of $\varphi_m$, it follows from
the definition \eqref{tran9a} of $v_m$ and estimate \eqref{tran16}
in Lemma \ref{tran15} that
\begin{equation}\label{tran30}
    \|H\|_{L^2(-\kappa_m,\kappa_m)}+ \|w\|_{L^2(Q_m)}\leq c.
\end{equation}
We have
\begin{equation*}
    \mathcal A_m=\sum\limits_{i+j=m-2, i,j\geq 0} a\partial_1^j
    H\partial_1^i\nabla w+ a\partial_1^m N\nabla v_m,
\end{equation*}
which yields
\begin{equation}\label{tran31}
    \|\mathcal A_m\|_{L^2(Q_m)}\leq\sum\limits_{i+j=m-2, i,j\geq 0} F_{ij}+
    \|\partial_1^m N\nabla v_m\|_{L^2(Q_m)},
\end{equation}
where
$$
F_{ij}=\|\partial_1^j H\, \partial_1^i\,w\|_{L^2(Q_m)}.
$$
Let us  estimate $F_{ij}$. We have
\begin{equation*}\begin{split}
F_{ij}^2=\int_{Q_m}|\partial_1^j H|^2\, |\partial_1^i w|^2\,
dy\leq\\ \Big\{\int_{-\kappa_m}^{\kappa_m}|\partial_1^j H|^2\,
dy_1\Big\}\, \Big\{\sup\limits_{y_1\in
(-\kappa_m,\kappa_m)}\,\int_{-\rho_m}^{\rho_m}|\partial_1^i
w(y_1,y_2)|^2\,dy_2\Big\}.
\end{split}\end{equation*}
Choose an arbitrary $ \sigma\in (1/2, 1)$. Recalling the definition
\eqref{anisotropic1} of norms $X^\infty$, $Y^j$ and inequality
\eqref{anisotropic4} we obtain
\begin{equation}\begin{split}\label{tran40}
F_{ij}\leq c \|H\|_{W^{j,2}(-\kappa_m, \kappa_m)}\, \|\partial_1^i
w\|_{X^\infty}\leq c \|H\|_{W^{j,2}(-\kappa_m,
\kappa_m)}\,\|\partial_1^i w\|_{Y^\sigma} .
\end{split}\end{equation}
Next, we employ the interpolation inequality in Sobolev space and
the special interpolation inequality \eqref{anisotropic5} to obtain
\begin{equation*}\begin{split}
\|H\|_{W^{j,2}(-\kappa_m, \kappa_m)}\leq c\|H\|_{L^2(-\kappa_m,
\kappa_m)}^{1-\frac{j}{m-1}}\,
\|H\|_{W^{m-1,2}(-\kappa_m,\kappa_m)}^{\frac{j}{m-1}}\leq c
\|H\|_{W^{m-1,2}(-\kappa_m,\kappa_m)}^{\frac{j}{m-1}}
\end{split}\end{equation*}
and
\begin{equation*}\begin{split}
\|\partial_1^iw\|_{Y^\sigma}\leq \|w\|_{Y^{i+\sigma}}\leq
c\|w\|_{L^2(Q_m)}^{1-\frac{i+\sigma}{m-1}}\,\,
\|w\|_{Y^{m-1}}^{\frac{i+\sigma}{m-1}}\leq c
\|w\|_{Y^{m-1}}^{\frac{i+\sigma}{m-1}}.
\end{split}\end{equation*}
Substituting these inequalities in \eqref{tran40}  we arrive at the
estimate
\begin{equation*}\begin{split}
F_{ij}\leq c \|H\|_{W^{m-1,2}(-\kappa_m,
\kappa_m)}^{\frac{j}{m-1}}\, \|
w\|_{Y^{m-1}}^{\frac{i+\sigma}{m-1}}.
\end{split}\end{equation*}
Notice that
$$
\frac{j}{m-1}+\frac{i+\sigma}{m-1}=\frac{j+i+\sigma}{m-1}=\frac{m-2+\sigma}{m
-1}=\lambda<1.
$$
Thus we get
\begin{equation*}\begin{split}
F_{ij}\leq c \big(\,\,\|H\|_{W^{m-1,2}(-\kappa_m,
\kappa_m)}^{\alpha}\, \| w\|_{Y^{m-1}}^{\beta}\,\,\big)^\lambda,
\end{split}\end{equation*}
where
$$
\alpha+\beta=\frac{1}{\lambda}(\frac{j}{m-1}+\frac{i+\sigma}{m-1})=1.
$$
Applying the Young inequality we obtain
\begin{equation}\label{tran32}
F_{ij}\leq c \big(\,\,\|H\|_{W^{m-1,2}(-\kappa_m,
\kappa_m)}^{\lambda}+ \| w\|_{Y^{m-1}}^{\lambda}\,\,\big).
\end{equation}
It remains to estimate the last term in inequality \eqref{tran31}.
By virtue of estimate \eqref{tran16} in Lemma \ref{tran15}, we have
\begin{equation*}\begin{split}
\|\partial_1^m N \nabla v_m\|_{L^2(Q_m)}\leq c \,\,\|\partial_1^m
N\|_{L^2(-\kappa_m, \kappa_m)}\,\, \| \nabla v_m\|_{X^\infty}\\\leq
c\|\partial_1^m N\|_{L^2(-\kappa_m, \kappa_m)}\,\, \| \nabla
v_m\|_{Y^1}\leq c\|\partial_1^m N\|_{L^2(-\kappa_m, \kappa_m)}.
\end{split}\end{equation*}
Substituting  this estimate and estimate \eqref{tran32} into
\eqref{tran31} we arrive at the inequality
\begin{equation}\label{tran36}
\|\mathcal A_m\|_{L^2(Q_m)}\leq c \|H\|_{W^{m-1,2}(-\kappa_m,
\kappa_m)}^{\lambda}\,+c\,\| w\|_{Y^{m-1}}^\lambda+c\|\partial_1^m
N\|_{L^2(-\kappa_m, \kappa_m)}.
\end{equation}
Since
$$w=\partial_1\nabla v_m\text{~~and~~}\|w\|_{Y^{m-1}}\leq
c(\|w\|_{L^2(Q_m)}+\|\partial_1^{m-1}w\|_{L^2(Q_m)},
$$
it follows from estimate \eqref{tran16} in Lemma \ref{tran15} that
\begin{equation}\label{tran38}
\|w\|_{Y^{m-1}}\leq c\big(\,1+\|\partial_1^{m}\nabla
v_m\|_{L^2(Q_m)}\,\big).
\end{equation}
On the other hand, the  equality $H=\partial_1N$ implies the
estimate
\begin{equation}\label{tran39}
\|H\|_{Y^{m-1}}\leq c\big(1+\|\partial_1^{m}N\|_{L^2(-\kappa_m,
\kappa_m)}\,\big).
\end{equation}
Substituting \eqref{tran38} and \eqref{tran39} into \eqref{tran36}
and noting that $0<\lambda<1$ we obtain desired inequality
\eqref{tran29}.

\subsection{The proof of estimate \eqref{tran41}  for $\mathcal
C_{m-1}$. } Recall that
$$ \mathcal C_{m-1}=a\nabla\varphi_m\cdot N\nabla v_{m-1},
$$
which yields
\begin{equation}\label{tran42}\begin{split}
\|\partial_1^{m-1}\mathcal C_{m-1}\|_{L^2(Q_m)}\leq
c\sum_{p=0}^{m-1}\|\partial_1^p\,(N\nabla
v_{m-1})\|_{L^2(Q_m)}\\\leq c+c\|\partial_1^{m-1}(N\nabla
v_{m-1})\|_{L^2(Q_m)}.
\end{split}\end{equation}
Notice that for $m=1$ estimate \eqref{tran41} is obviously true.
Next, it follows from \eqref{tran42} that
\begin{equation*}\begin{split}
\|\partial_1\mathcal C_1\|_{L^2(Q_1)}\leq c+c\|\partial_1(N\nabla
v_1)\|_{L^2(Q_m)}\leq\\
c+c\|N \partial_1\nabla v_1\|_{L^2(Q_1)}+\|\partial_1 N\nabla
v_1\|_{L^2(Q_1)}.
\end{split}\end{equation*}
From this, estimate \eqref{tran16} in Lemma \ref{tran15}, and
inequality \eqref{tran5} we conclude that
\begin{equation*}\begin{split}
\|\partial_1\mathcal C_1\|_{L^2(Q_1)}\leq c+c\|\partial_1(N\nabla
v_1)\|_{L^2(Q_m)}\leq\\
c+\|\partial_1 N\|_{L^2(-\kappa_m,\kappa_m)}\|\nabla
v_1\|_{X^\infty}\leq c +c\|\nabla v_1\|_{X^\infty}.
\end{split}\end{equation*}
Employing  estimates \eqref{anisotropic4} and \eqref{tran16} we
finally obtain
\begin{equation*}\begin{split}
\|\partial_1\mathcal C_1\|_{L^2(Q_1)}\leq c +c\|\nabla
v_1\|_{Y^1}\leq c+\|\partial_1 \nabla v_1\|_{L^2(Q_m)}\leq c.
\end{split}\end{equation*}
Hence estimate \eqref{tran41} holds true for $m=1,2$. Let us
consider the case $m\geq 3$. By virtue of \eqref{tran42}, we have
\begin{equation}\label{tran43}\begin{split}
\|\partial_1^{m-1}\mathcal C_{m-1}\|_{L^2(Q_m)} \leq
c+c\|\partial_1^{m-1}(N\nabla v_{m-1})\|_{L^2(Q_m)}\leq\\
c+c\|\partial_1^{m-1} \nabla v_{m-1}\|_{L^2(Q_m)}+\sum_{i+j=m-1,
j\geq 1} \|\partial_1^j N\,\partial_1^i\nabla v_{m-1}\|_{L^2(Q_m)}.
\end{split}\end{equation}
Arguing as before we obtain
\begin{equation*}
 \|\partial_1^j N\,\partial_1^i\nabla v_{m-1}\|_{L^2(Q_m)}\leq
 c\|N\|_{W^{j,2}(-\kappa_m,\kappa_m)}\,\,\|\partial_1^i\nabla v_{m-1} \|_{X^\infty}.
\end{equation*}
Fix an arbitrary $\sigma\in (1/2,1)$. Estimate \eqref{anisotropic4}
implies
$$
\|\partial_1^i\nabla v_{m-1} \|_{X^\infty}\leq c\|\partial_1^i\nabla
v_{m-1}\|_{Y^\sigma}\leq c\|\nabla v_{m-1}\|_{Y^{i+\sigma}}.
$$
Thus we get
\begin{equation}\label{tran45}
 \|\partial_1^j N\,\partial_1^i\nabla v_{m-1}\|_{L^2(Q_m)}\leq
 c\|N\|_{W^{j,2}(-\kappa_m,\kappa_m)}\,\,\|\nabla v_{m-1} \|_{Y^{i+1}}.
\end{equation}
It follows from \eqref{tran5} and the interpolation inequality that
\begin{multline}\label{tran46}
\|N\|_{W^{j,2}(-\kappa_m,\kappa_m)}\leq
\|N\|_{W^{1,2}(-\kappa_m,\kappa_m)}^{\frac{m-1-j}{m-2}}
\|N\|_{W^{m-1,2}(-\kappa_m,\kappa_m)}^{\frac{j-1}{m-2}} \\\leq c
\|N\|_{W^{m-1,2}(-\kappa_m,\kappa_m)}^{\frac{j-1}{m-2}}.
\end{multline}
On the other hand, interpolation inequality \eqref{anisotropic5} and
estimate \eqref{tran16} imply
\begin{equation}\label{tran47}\begin{split}
\|\nabla v_{m-1} \|_{Y^{i+1}}\leq c \|\nabla v_{m-1}
\|_{Y^{1}}^{1-\frac{i}{m-2}}\|\nabla v_{m-1}\|_{Y^{m-1}}^{\frac{i}{m-2}}\leq \\
c\|\nabla v_{m-1}\|_{Y^{m-1}}^{\frac{i}{m-2}}\leq
c(1+\|\partial_1^{m-1}\nabla v_{m-1}\|_{L^2(Q_m)})^{\frac{i}{m-2}}.
\end{split}\end{equation}
Substituting \eqref{tran46}-\eqref{tran47} into \eqref{tran45} and
noting that
\begin{multline*}
\|N\|_{W^{m-1,2}(-\kappa_m,\kappa_m)}\leq
c(\|N\|_{L^2(-\kappa_m,\kappa_m)}+\|\partial_1^{m-1}N\|_{L^2(-\kappa_m,\kappa_m)})\\
\leq c(1+\|\partial_1^{m-1}N\|_{L^2(-\kappa_m,\kappa_m)})
\end{multline*}
we arrive at the estimate
\begin{equation*}\begin{split}
 \|\partial_1^j N\,\partial_1^i\nabla v_{m-1}\|_{L^2(Q_m)}\leq\\
 c(1+\|\partial_1^{m-1}N\|_{L^2(-\kappa_m,\kappa_m)})^{\frac{j-1}{m-2}}
\,(1+\|\partial_1^{m-1}\nabla v_{m-1}\|_{L^2(Q_m)})^{\frac{i}{m-2}}.
\end{split}\end{equation*}
Since
$$
\frac{j-1}{m-2}+\frac{i}{m-2}=\frac{i+j-1}{m-2}=1,
$$
we can apply the Young inequality to obtain
\begin{equation*}\begin{split}
 \|\partial_1^j N\,\partial_1^i\nabla v_{m-1}\|_{L^2(Q_m)}\leq\\
 c\big(\,1+\|\partial_1^{m-1}N\|_{L^2(-\kappa_m,\kappa_m)}+
 \|\partial_1^{m-1}\nabla v_{m-1}\|_{L^2(Q_m)}\,\big).
\end{split}\end{equation*}
Substituting this inequality into \eqref{tran43} we finally obtain
desired estimate \eqref{tran41}

\subsection{The proof of estimate \eqref{tran58} for $\mathcal
B_{m-1}$.} Since $a$ and $N$ are independent of $y_1$, it follows
from the expression \eqref{tran10} for $\mathcal B_1$ that
\begin{multline}\label{tran52}
\|\partial_1^{m}\mathcal B_{m-1}\|_{L^2(Q_m)}\leq
c\|\partial_1^m(v_{m-1}\nabla\varphi_m)\|_{L^2(Q_m)}\\+c\sum_{i+j=m,
i\leq m-1}\|\partial_1^j N\,
\partial_1^i(v_{m-1}\nabla\varphi_{m}\|_{L^2(Q_m)}.
\end{multline}
We have
\begin{equation*}
    \|\partial_1^m(v_{m-1}\nabla\varphi_m)\|_{L^2(Q_m)}\leq
    c\sum_{0\leq p\leq m}\|\partial_1^p(v_{m-1}\nabla\varphi_m)\|_{L^2(Q_{m-1})}
\end{equation*}
which along with the Poincare inequality yields the estimate
\begin{equation}\label{tran53}\begin{split}
    \|\partial_1^m(v_{m-1}\nabla\varphi_m)\|_{L^2(Q_m)}\leq
    c(1+\|\partial_1^m v_{m-1}\|_{L^2(Q_{m-1})})\\\leq
     c(1+\|\partial_1^{m-1}\nabla v_{m-1}\|_{L^2(Q_{m-1})}).
\end{split}\end{equation} For  nonnegative integers $i,j$, satisfying
relations $i+j=m$, $0\leq i\leq m-1$, we have
\begin{equation}\begin{split}\label{tran54}
    \|\partial_1^j N\,\partial_1^i(v_{m-1}\nabla\varphi_m)\|_{L^2(Q_m)}\leq\\
    c\|\partial_1^j N\|_{L^2(-\kappa_m,\kappa_m)}\,
    \| \partial_1^i(v_{m-1}\nabla\varphi_m)\|_{X^\infty}.
\end{split}\end{equation} Recall definitions \eqref{anisotropic1} and
\eqref{anisotropic3} of the Banach spaces $X^\infty$ and $Y^s$. Now
choose an arbitrary $\sigma\in (1/2, 1)$. By the embedding
inequality \eqref{anisotropic5}, we have
$$
\| \partial_1^i(v_{m-1}\nabla\varphi_m)\|_{X^\infty}\leq c\|
v_{m-1}\nabla\varphi_m\|_{Y^{i+\sigma}},
$$
which along with \eqref{tran54} yields
\begin{equation}\begin{split}\label{tran55}
    \|\partial_1^j N\,\,\partial_1^i(v_{m-1}\nabla\varphi_m)\|_{L^2(Q_m)}\leq
    c\|\partial_1^j N\|_{L^2(-\kappa_m,\kappa_m)}\,\,
    \|v_{m-1}\nabla\varphi_m\|_{Y^{i+\sigma}}.
\end{split}\end{equation}
Next notice that by virtue of estimates \eqref{tran5} and
\eqref{tran16}, we have
$$
\|N\|_{W^{1,2}(-\kappa_m,\kappa_m)}+
\|v_{m-1}\nabla\varphi_n\|_{Y^1}\leq c.
$$
From this and the interpolation inequality we obtain
$$
\|\partial_1^j N\|_{L^2(-\kappa_m,\kappa_m)}\leq
c\|N\|_{W^{m,2}(-\kappa_m, \kappa_m)}^{\frac{j-1}{m-1}},\quad
\|v_{m-1}\nabla\varphi_m\|_{Y^{i+\sigma}}\leq
c\|v_{m-1}\nabla\varphi_m\|_{Y^m}^{\frac{i+\sigma-1}{m-1}}.
$$
Since
$$
\frac{j-1}{m-1}+\frac{i+\sigma-1}{m-1}=1-\frac{1-\sigma}{m-1}\leq 1,
$$
it follows from \eqref{tran55} and the Young inequality that
\begin{equation}\begin{split}\label{tran56}
    \|\partial_1^j
    N\,\,\partial_1^i(v_{m-1}\nabla\varphi_m)\|_{L^2(Q_m)}\leq\\
    c(1+\|N\|_{W^{m,2}(-\kappa_m, \kappa_m)}+\|v_{m-1}\nabla\varphi_m\|_{Y^m}).
\end{split}\end{equation}
Notice that
\begin{equation*}
\|N\|_{W^{m,2}(-\kappa_m, \kappa_m)}\leq c(1+\|\partial_1^m
N\|_{L^2(-\kappa_m, \kappa_m)}).
\end{equation*}
Arguing as in the proof of \eqref{tran53} we obtain
\begin{equation*}\begin{split}
\|v_{m-1}\nabla\varphi_m\|_{Y^m}\leq c(1+\|\partial_1^m
(v_{m-1}\nabla\varphi_m)\|_{L^2(Q_m)}\\
\leq c(1+\|\partial_1^{m-1}\nabla v_{m-1}\|_{L^2(Q_{m-1})}\,).
\end{split}\end{equation*}
Substituting these inequalities  into \eqref{tran56} we arrive at
the estimate
\begin{equation*}\begin{split}
    \|\partial_1^j N\,\,\partial_1^i(v_{m-1}\nabla\varphi_m)\|_{L^2(Q_m)}\leq
    c(1+\|\partial_1^m
N\|_{L^2(-\kappa_m, \kappa_m)}+\|\partial_1^{m-1}\nabla
v_{m-1}\|_{L^2(Q_{m-1})}\,).
\end{split}\end{equation*}
Combining this estimate with estimates \eqref{tran53} and
\eqref{tran52} we arrive at the desired estimate \eqref{tran58} for
$\mathcal B_{m-1}$.

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\section{Proof of Lemma \ref{appb38}}\label{APPE}
\renewcommand{\theequation}{12.\arabic{equation}}
\setcounter{equation}{0}


Introduce the denotations
\begin{equation}\label{appb40}
\Phi=\varphi\, n, \quad k=K\, n, \quad \varphi=\Phi\cdot n, \quad
K=k\cdot n,
\end{equation}
where $K$ is the scalar curvature.    We have
\begin{equation*}
\partial_s^r\Phi-\nabla_s^r\Phi=\partial_s^r(\varphi\,
n)-\partial_s^r\varphi\,
n=\sum_{i=0}^{r-1}c_i\partial_s^i\varphi\,\partial_s^{r-i}n.
\end{equation*}
Note that
$$
\partial_s n=\partial_s \tau^\bot=k^\bot, \quad
k^\bot=(-k_2,k_1)=-K\tau.
$$
It follows that
\begin{equation}\label{appb41}
\partial_s^r\Phi-\nabla_s^r\Phi=\sum_{i=0}^{r-1}c_i\partial_s^i\varphi\,\partial_s^{r-1-i}k^\bot.
\end{equation}
Let us estimate $L^2$-norm of every term in the right hand side.
First we consider the case when
$$
i\geq 1,\quad j=r-1-i\geq 1.
$$
The H\"{o}lder inequality implies the estimate
$$
\|\partial_s^i\varphi\partial_s^j
k\|_{H^0_\sharp}^2=\int_{\Gamma}|\partial_s^i\varphi|^2|\partial_s^j
k|^2\,ds\leq \Big(\int_\Gamma |\partial_s^i
\varphi|^{\frac{2(r-1)}{i}}\,ds\Big)^{\frac{i}{r-1}}\,\Big(\int_\Gamma
|\partial_s^j k|^{\frac{2(r-1)}{j}}\,ds\Big)^{\frac{j}{r-1}},
$$
which can be written in the equivalent form
\begin{equation*}
\|\partial_s^i\varphi\partial_s^j k\|_{H^0_\sharp}\leq
\|\partial_s^i\varphi \|_{L^{\frac{2(r-1)}{i}}(0,\mathcal L)}\,
\|\partial_s^j k \|_{L^{\frac{2(r-1)}{j}}(0,\mathcal L)}.
\end{equation*}
From this and the the Gagliardo-Nirenberg inequality \eqref{appb10}
we obtain
\begin{equation}\label{appb42}
\|\partial_s^i\varphi\partial_s^j k\|_{L^2(0, \mathcal L)}\leq c
(\|\varphi\|_{L^\infty(0,\mathcal L)}^{1-\frac{i}{r-1}}
\|k\|_{L^\infty(0,\mathcal L)}^{1-\frac{j}{r-1}})\,\,(
\|\varphi\|_{H^{r-1}_\sharp}^{\frac{i}{r-1}}
\|k\|_{H^{r-1}_\sharp}^{\frac{j}{r-1}}).
\end{equation}
It follows from the embedding inequality \eqref{appb8}  that for
$\sigma\in (1/2,1)$, we have
$$
\|\varphi\|_{L^\infty(0,\mathcal L)}\leq
c\|\varphi\|_{H^\sigma_\sharp}.
$$
Next, the interpolation inequality gives
$$
\|\varphi\|_{H^\sigma_\sharp}\leq
\|\varphi\|_{H^0_\sharp}^{1-\frac{\sigma}{r}}\,\,\|\varphi\|_{H^r_\sharp}^{\frac{\sigma}{r}},
\quad \|\varphi\|_{H^{r-1}_\sharp}\leq
\|\varphi\|_{H^0_\sharp}^{\frac{1}{r}}\,\,\|\varphi\|_{H^r_\sharp}^{1-\frac{1}{r}},
$$
which yields
\begin{equation*}\begin{split}
\|\varphi\|_{L^\infty(0,\mathcal
L)}^{1-\frac{i}{r-1}}\,\,\|\varphi\|_{H^{r-1}_\sharp}^{\frac{i}{r-1}}\leq
c\|\varphi\|_{H^0_\sharp}^{\alpha_i}\,\,\|\varphi\|_{H^{r}_\sharp}^{\beta_i},
\end{split}\end{equation*}
where
\begin{equation}\label{appb45}
\alpha_i=(1-\frac{\sigma}{r})(1-\frac{i}{r-1})+\frac{i}{r(r-1)},
\quad \beta_i=\frac{\sigma}{r}(1-\frac{i}{r-1})\,\,
+\frac{r-1}{r}\frac{i}{r-1}.
\end{equation}
Repeating these arguments we obtain
\begin{equation*}\begin{split}
\|k\|_{L^\infty(0,\mathcal
L)}^{1-\frac{j}{r-1}}\,\,\|k\|_{H^{r-1}_\sharp}^{\frac{j}{r-1}}\leq
c\|k\|_{H^0_\sharp}^{\alpha_j}\,\,\|k\|_{H^{r}_\sharp}^{\beta_j}.
\end{split}\end{equation*}
The quantities $\alpha_j$ and $\beta_j$ are given by the formulae
\eqref{appb45}  with $i$ replaced by $j$. From this and
\eqref{appb42} we conclude that
\begin{equation}\label{appb47}
\|\partial_s^i\varphi\,\,\partial_s^j k\|_{H^0_\sharp}\leq c\big(
\|\varphi\|_{H^0_\sharp}^{\alpha_i}\,\|k\|_{H^0_\sharp}^{\alpha_j}\big)\,\,\big(
\|\varphi\|_{H^{r}_\sharp}^{\beta_i}\,\|k\|_{H^{r}_\sharp}^{\beta_j}\big).
\end{equation}
Since $i+j=r-1$ we have
$$
\alpha_i+\alpha_j=1+\frac{1-\sigma}{r}, \quad
\beta_i+\beta_j=1-\frac{1-\sigma}{r}.
$$
In other words, we have
$$
\alpha_i=(1+\frac{1-\sigma}{r})\alpha_i^*,\,\,\,\alpha_j=(1+\frac{1-\sigma}{r})\alpha_j^*
, \quad \beta_i=(1-\frac{1-\sigma}{r})\beta_i^*,
\,\,\beta_j=(1-\frac{1-\sigma}{r})\beta_j^*,
$$
where $\alpha_i^*+\alpha_j^*=1$ and $\beta_i^*+\beta_j^*=1$. From
this and the Young inequality we obtain
\begin{equation*}\begin{split}
\|\varphi\|_{H^0_\sharp}^{\alpha_i}\,\|k\|_{H^0_\sharp}^{\alpha_j}\leq
(\|\varphi\|_{H60_\sharp}+\|k\|_{H^0_\sharp})^{1+\frac{1-\sigma}{r}}, \\
\|\varphi\|_{H^r_\sharp}^{\beta_i}\,\|k\|_{H^r_\sharp}^{\beta_j}\leq
(\|\varphi\|_{H^r_\sharp}+\|k\|_{H^r_\sharp})^{1-\frac{1-\sigma}{r}}.
\end{split}\end{equation*}
Combining these estimates with \eqref{appb47} we conclude that the
inequality
\begin{equation}\label{appb48}
\|\partial_s^i\varphi\,\,\partial_s^j k\|_{H^0_\sharp}\leq
\\c (\|\varphi\|_{H^0_\sharp}+\|k\|_{H^0_\sharp})^{1+\frac{1-\sigma}{r}}\,\,
(\|\varphi\|_{H^r_\sharp}+\|k\|_{H^r_\sharp})^{1-\frac{1-\sigma}{r}}
\end{equation}
holds for every $1\leq i\leq r-2$ and $j=r-1-i$. It remains to
consider the cases $i=0$ and $i=r-1$. We have for $\sigma\in
(1/2,1)$,
\begin{equation*}
\|\varphi\,\,\partial_s^{r-1} k\|_{H^0_\sharp}+
\|\partial_s^{r-1}\varphi\,\, k\|_{H^0_\sharp}\leq c
\|\varphi\|_{H^\sigma_\sharp}\,\|k\|_{H^{r-1_\sharp}}+
c\|k\|_{H^\sigma_\sharp}\,\|\varphi\|_{H^{r-1}_\sharp}.
\end{equation*}
Next, the interpolation inequality implies
\begin{equation*}\begin{split}
\|\varphi\|_{H^\sigma_\sharp}\leq
c\|\varphi\|_{H^0_\sharp}^{1-\frac{\sigma}{r}}\,\|\varphi\|_{H^r_\sharp}^{\frac{\sigma}{r}},
\quad  \|\varphi\|_{H^{r-1}_\sharp}\leq
c\|\varphi\|_{H^0_\sharp}^{\frac{1}{r}}\,\|\varphi\|_{H^r_\sharp}^{1-\frac{1}{r}}.
\end{split}\end{equation*}
The similar estimates holds true for $k$. Thus we get
\begin{equation*}\begin{split}
\|\varphi\,\,\partial_s^{r-1} k\|_{H^0_\sharp}+
\|\partial_s^{r-1}\varphi\,\, k\|_{H^0_\sharp}\leq\\
 c(\|k\|_{H^0_\sharp}^{1-\frac{\sigma}{r}}\|\varphi\|_{H^0_\sharp}^{\frac{1}{r}})
 \,\,(\|k\|_{H^r}^{\frac{\sigma}{r}}\,
\|\varphi\|_{H^r}^{1-\frac{1}{r}})+
\\
 c(\|\varphi\|_{H^0_\sharp}^{1-\frac{\sigma}{r}}\|k\|_{H^0_\sharp}^{\frac{1}{r}})
 \,\,(\|\varphi\|_{H^r}^{\frac{\sigma}{r}}\,
\|k\|_{H^r}^{1-\frac{1}{r}}) .
\end{split}\end{equation*}
It follows that
\begin{equation*}\begin{split}
\|\varphi\,\,\partial_s^{r-1} k\|_{H^0_\sharp}+
\|\partial_s^{r-1}\varphi\,\, k\|_{H^0_\sharp}\leq\\
 c\,\big(\|k\|_{H^0_\sharp}^{1-\frac{\sigma}{r}}\,\,\|\varphi\|_{H^0_\sharp}^{\frac{1}{r}}
 +\|\varphi\|_{H^0_\sharp}^{1-\frac{\sigma}{r}}\,\,\|k\|_{H^0_\sharp}^{\frac{1}{r}}\,\big)\,\,\,
\big(\|k\|_{H^r_\sharp}^{1-\frac{\sigma}{r}}\,\,
\|\varphi\|_{H^r_\sharp}^{1-\frac{1}{r}}+
\|\varphi\|_{H^r_\sharp}^{1-\frac{\sigma}{r}}\,\,
\|k\|_{H^r_\sharp}^{1-\frac{1}{r}}\big) .
\end{split}\end{equation*}
Using the simple inequality
$$
a^\lambda b^\mu+a^\mu b^\lambda \leq 2 a^{\lambda+\mu}+2 b
^{\lambda+\mu}, \quad a,b,\lambda, \mu\geq 0,
$$
we finally obtain the estimate
\begin{equation*}\begin{split}
\|\varphi\,\,\partial_s^{r-1} k\|_{H^0_\sharp}+
\|\partial_s^{r-1}\varphi\,\, k\|_{H^0_\sharp}\leq\\
 c\,\big(\,\|k\|_{H^0_\sharp}^{1+\frac{1-\sigma}{r}}+\|\varphi\|_{H^0_\sharp}^{1+\frac{1-\sigma}{r}}\,\big)\,\,
\big(\,\|k\|_{H^r_\sharp}^{1-\frac{1-\sigma}{r}}+
\|\varphi\|_{H^r_\sharp}^{1-\frac{1-\sigma}{r}}\big).
\end{split}\end{equation*}
It follows from this estimate and estimate   \eqref{appb48} that the
inequality
\begin{equation*}
\|\partial_s^i\varphi\,\,\partial_s^j k\|_{H^0_\sharp}\leq
\\c \big(\,\|\varphi\|_{H^0_\sharp}+\|k\|_{H^0_\sharp}\,\big)^{1+\frac{1-\sigma}{r}}\,\,
\big(\,\|\varphi\|_{H^r_\sharp}+\|k\|_{H^r_\sharp}\,\big)^{1-\frac{1-\sigma}{r}}
\end{equation*}
holds for every $0\leq i\leq r-1$ and $j=r-1-i$.
 Combining this result with identity
\eqref{appb41} we obtain desired estimate \eqref{appb39}.














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