text:
\begin{document}
\def\qed{\hfill$\Box$}
\newtheorem{theorem}{Theorem}{\it}{\rm}
\newtheorem{corollary}{Corollary}{\it}{\rm}
\newtheorem{lemma}{Lemma}{\it}{\rm}
\newtheorem{proposition}{Proposition}{\it}{\rm}
\makeatletter
 \@addtoreset{corollary}{section}
 \@addtoreset{equation}{section}
 \@addtoreset{examples}{section}
 \@addtoreset{lemma}{section}
 \@addtoreset{proposition}{section}
 \@addtoreset{remarks}{section}
 \@addtoreset{theorem}{section}
\makeatother
\theoremstyle{definition}
\newtheorem{examples}{Examples}
\newtheorem{remark}{Remark}
\newtheorem{remarks}{Remarks}
\renewcommand{\thecorollary}{\arabic{section}.\arabic{corollary}}
\renewcommand{\theequation}{\arabic{section}.\arabic{equation}}
\renewcommand{\theexamples}{\arabic{section}.\arabic{examples}}
\renewcommand{\thelemma}{\arabic{section}.\arabic{lemma}}
\renewcommand{\theproposition}{\arabic{section}.\arabic{proposition}}
\renewcommand{\theremark}{\arabic{section}.\arabic{remark}}
\renewcommand{\theremarks}{\arabic{section}.\arabic{remarks}}
\renewcommand{\thetheorem}{\arabic{section}.\arabic{theorem}}
\makeatletter
\def\@thm#1#2#3{
  \ifhmode\unskip\unskip\par\fi
  \normalfont
  \trivlist
  \let\thmheadnl\relax
  \let\thm@swap\@gobble
  \thm@notefont{\fontseries\mddefault\upshape}
  \thm@headpunct{}
  \thm@headsep 5\p@ plus\p@ minus\p@\relax
  \thm@space@setup
  #1
  \@topsep \thm@preskip
  \@topsepadd \thm@postskip
  \def\@tempa{#2}\ifx\@empty\@tempa
    \def\@tempa{\@oparg{\@begintheorem{#3}{}}[]}
  \else
    \refstepcounter{#2}
    \def\@tempa{\@oparg{\@begintheorem{#3}{\csname the#2\endcsname}}[]}
  \fi
  \@tempa
}
\renewenvironment{proof}[1][\proofname]{\par
  \pushQED{\qed}
  \normalfont \topsep6\p@\@plus6\p@\relax
  \trivlist
  \item[\hskip\labelsep
        \itshape
    #1\@addpunct{}]\ignorespaces
}{
  \popQED\endtrivlist\@endpefalse
}
\providecommand{\proofname}{Proof}
\makeatother
 \newcommand*{\al}{\alpha}
 \newcommand*{\ba}{\beta}
 \newcommand*{\Da}{\Delta}
 \newcommand*{\da}{\delta}
 \newcommand*{\Ga}{\Gamma}
 \newcommand*{\ga}{\gamma}
 \newcommand*{\ia}{\iota}
 \newcommand*{\ka}{\kappa}
 \newcommand*{\wk}{{\wt{\ka}}}
 \newcommand*{\wkk}{{\wt{\ka}\ka}}
 \newcommand*{\kwk}{{\ka\wt{\ka}}}
 \newcommand*{\lda}{\lambda}
 \newcommand*{\na}{\nabla}
 \newcommand*{\Om}{\Omega}
 \newcommand*{\om}{\omega}
 \newcommand*{\oO}{\ol{\Omega}}
 \newcommand*{\Sa}{\Sigma}
 \newcommand*{\sa}{\sigma}
 \newcommand*{\ta}{\theta}
 \newcommand*{\za}{\zeta}
 \newcommand*{\ve}{\varepsilon}
 \newcommand*{\vp}{\varphi}
 \newcommand*{\vt}{\vartheta}
 \newcommand*{\BA}{{\mathbb A}}
 \newcommand*{\BB}{{\mathbb B}}
 \newcommand*{\BC}{{\mathbb C}}
 \newcommand*{\BE}{{\mathbb E}}
 \newcommand*{\BF}{{\mathbb F}}
 \newcommand*{\BG}{{\mathbb G}}
 \newcommand*{\BH}{{\mathbb H}}
 \newcommand*{\BL}{{\mathbb L}}
 \newcommand*{\BN}{{\mathbb N}}
 \newcommand*{\BR}{{\mathbb R}}
 \newcommand*{\BX}{{\mathbb X}}
 \newcommand*{\cA}{{\mathcal A}}
 \newcommand*{\cB}{{\mathcal B}}
 \newcommand*{\cC}{{\mathcal C}}
 \newcommand*{\cD}{{\mathcal D}}
 \newcommand*{\cL}{{\mathcal L}}
 \newcommand*{\cR}{{\mathcal R}}
 \newcommand*{\cW}{{\mathcal W}}
 \newcommand*{\cX}{{\mathcal X}}
 \newcommand*{\cY}{{\mathcal Y}}
 \newcommand*{\gA}{{\mathfrak A}}
 \newcommand*{\gsa}{{\mathfrak a}}
 \newcommand*{\gB}{{\mathfrak B}}
 \newcommand*{\gF}{{\mathfrak F}}
 \newcommand*{\gK}{{\mathfrak K}}
 \newcommand*{\gM}{{\mathfrak M}}
 \newcommand*{\gN}{{\mathfrak N}}
 \newcommand*{\gss}{{\mathfrak s}}
 \newcommand*{\gsu}{{\mathfrak u}}
 \newcommand*{\gsv}{{\mathfrak v}}
 \newcommand*{\gW}{{\mathfrak W}}
 \newcommand*{\sA}{{\mathsf A}}
 \newcommand*{\sBC}{{\mathsf{BC}}}
 \newcommand*{\sW}{{\mathsf W}}
 \newcommand*{\bal}{\begin{aligned}}
 \newcommand*{\eal}{\end{aligned}}
 \newcommand*{\uti}[1]{(#1)\space\space}
 \newcommand*{\qa}{,\qquad}
 \newcommand*{\qb}{,\quad}
 \newcommand*{\mf}[1]{\boldsymbol{#1}}
 \newcommand*{\ci}{\mathaccent"7017 }
 \newcommand*{\hb}[1]{\hbox{$#1$}}

 \newcommand*{\sdot}{\!\cdot\!}
 \newcommand*{\sco}{\kern2pt\colon\kern2pt}
 \newcommand*{\sn}{\kern1pt|\kern1pt}
 \newcommand*{\bsn}{\kern1pt\big|\kern1pt}
 \newcommand*{\Bsn}{\kern1pt\Big|\kern1pt}
 \newcommand*{\ssm}{\!\setminus\!}
 \newcommand*{\eesdot}{\hbox{$[\![{}\sdot{}]\!]$}}
 \newcommand*{\eea}[1]{\hbox{$[\![#1]\!]$}}
 \newcommand*{\beea}[1]{\hbox{$\big[\!\kern-1pt\big[#1\big]\!\kern-1pt\big]$}}
 \newcommand*{\Beea}[1]{\hbox{$\Big[\!\kern-1pt\Big[#1\Big]\!\kern-1pt\Big]$}}
 \newcommand*{\vsdot}{\hbox{$\vert\sdot\vert$}}
 \newcommand*{\Vsdot}{\hbox{$\Vert\sdot\Vert$}}
 \newcommand*{\npbd}{\postdisplaypenalty=20000}
 \newcommand*{\po}{\pagebreak[2] }
 \newcommand*{\prsn}{\hbox{$(\cdot\sn\cdot)$}}
 \newcommand*{\pw}{\hbox{$\dl{}\sdot{},{}\sdot{}\dr$}}
 \newcommand*{\mfA}{{\mf{A}}}
 \newcommand*{\mfsA}{{\mf{\sA}}}
 \newcommand*{\mfB}{{\mf{B}}}
 \newcommand*{\mfC}{{\mf{C}}}
 \newcommand*{\mfE}{{\mf{E}}}
 \newcommand*{\mff}{{\mf{f}}}
 \newcommand*{\mfg}{{\mf{g}}}
 \newcommand*{\mfh}{{\mf{h}}}
 \newcommand*{\mfu}{{\mf{u}}}
 \newcommand*{\mfv}{{\mf{v}}}
 \newcommand*{\mfW}{{\mf{W}}}
 \newcommand*{\mfsW}{{\mf{\sW}}}
 \newcommand*{\mfw}{{\mf{w}}}
 \newcommand*{\mfz}{{\mf{z}}}
 \newcommand*{\mfvp}{{\mf{\vp}}}
 \newcommand*{\mfpsi}{{\mf{\psi}}}
 \newcommand*{\ol}{\overline}
 \newcommand*{\ul}{\underline}
 \newcommand*{\wh}{\widehat}
 \newcommand*{\wt}{\widetilde}
 \newsavebox{\Prel}
 \sbox{\Prel}{\begin{picture}(2,6)(0,0)\put(1,2.5){\circle*{2}}\end{picture}}
 \newcommand*{\btdot}{\mathrel{\usebox{\Prel}}}
 \newcommand*{\fdot}{{\usebox{\Prel}}}
 \newsavebox{\supiPrel}
 \sbox{\supiPrel}{\begin{picture}(2,2)(0,0)
 \put(.9,.9){\circle*{1}}
 \put(1.1,.9){\circle*{1}}
 \put(1,.8){\circle*{1}}
 \put(1,1){\circle*{1}}
 \end{picture}}
 \newcommand*{\ifdot}{{\usebox{\supiPrel}}}
 \newcommand*{\vph}{\vphantom}
 \newcommand*{\hr}{\hookrightarrow}
 \newcommand*{\ra}{\rightarrow}
 \newcommand*{\dl}{\langle}
 \newcommand*{\dr}{\rangle}
 \newcommand*{\card}{\mathop{\rm card}\nolimits}
 \newcommand*{\diag}{{\mathop{\rm diag}\nolimits}}
 \newcommand*{\diam}{\mathop{\rm diam}\nolimits}
 \newcommand*{\tdiv}{\mathop{\rm div}\nolimits}
 \newcommand*{\grad}{\mathop{\rm grad}\nolimits}
 \newcommand*{\graph}{\mathop{\rm graph}\nolimits}
 \newcommand*{\id}{{\rm id}}
 \newcommand*{\supp}{\mathop{\rm supp}\nolimits}
 \newcommand*{\vol}{\mathop{\rm vol}\nolimits}
 \newcommand*{\Lis}{{\mathcal L}{\rm is}}
 \newcommand*{\loc}{{\rm loc}}
 \renewcommand*{\Re}{\mathop{\rm Re}\nolimits}
 \newcommand*{\imi}{{\mit i\kern1pt}}
 \newcommand*{\mi}[1]{{#1\textrm-}}
 \newcommand*{\is}{\subset}
 \newcommand*{\bt}{\bullet}
 \newcommand*{\es}{\emptyset}
 \newcommand*{\iy}{\infty}
 \newcommand*{\mt}{\mapsto}
 \newcommand*{\nat}{\natural}
 \newcommand*{\nag}{\na_{\cona\cona g}}
 \newcommand*{\naka}{\na_{\cona\ka}}
 \newcommand*{\naSa}{\na_{\cona\Sa}}
 \newcommand*{\pl}{\partial}
 \newcommand*{\pa}{\partial^\alpha}
 \newcommand*{\coc}{\kern.75pt}
 \newcommand*{\cona}{\kern-1pt}
 \newcommand*{\coU}{\kern-1pt}
 \newcommand*{\coW}{\kern-1pt}
 \newcommand*{\AB}{(\cA,\cB)}
 \newcommand*{\ArBr}{(\cA_r,\cB_r)}
 \newcommand*{\EE}{(E,E)}
 \newcommand*{\EF}{(E,F)}
 \newcommand*{\Hmgm}{(\BH^m,g_m)}
 \newcommand*{\Mg}{(M,g)}
 \newcommand*{\Migi}{(M_i,g_i)}
 \newcommand*{\whMwhg}{(\widehat M,\widehat g)}
 \newcommand*{\MV}{(M,V)}
 \newcommand*{\Nh}{(N,h)}
 \newcommand*{\oOgm}{(\oO,g_m)}
 \newcommand*{\RC}{(\BR,\BC)}
 \newcommand*{\Rmgm}{(\BR^m,g_m)}
 \newcommand*{\cRcRc}{(\cR,\cR^c)}
 \newcommand*{\Vg}{(V,g)}
 \newcommand*{\BXF}{(\BX,F)}
 \newcommand*{\BUC}{BU\kern-.3ex C}

 \makeatletter
 \newif\ifinany@
 \newcount\column@
 \def\column@plus{
    \global\advance\column@\@ne
 }
 \newcount\maxfields@
 \def\add@amps#1{
    \begingroup
        \count@#1
        \DN@{}
        \loop
            \ifnum\count@>\column@
                \edef\next@{&\next@}
                \advance\count@\m@ne
        \repeat
    \@xp\endgroup
    \next@
 }
 \def\Let@{\let\\\math@cr}
 \def\restore@math@cr{\def\math@cr@@@{\cr}}
 \restore@math@cr
 \def\default@tag{\let\tag\dft@tag}
 \default@tag
 \newbox\strutbox@
 \def\strut@{\copy\strutbox@}
 \addto@hook\every@math@size{
  \global\setbox\strutbox@\hbox{\lower.5\normallineskiplimit
         \vbox{\kern-\normallineskiplimit\copy\strutbox}}}
 \renewcommand{\start@aligned}[2]{
    \RIfM@\else
        \nonmatherr@{\begin{\@currenvir}}
    \fi
    \null\,
    \if #1t\vtop \else \if#1b \vbox \else \vcenter \fi \fi \bgroup
        \maxfields@#2\relax
        \ifnum\maxfields@>\m@ne
            \multiply\maxfields@\tw@
            \let\math@cr@@@\math@cr@@@alignedat
        \else
            \restore@math@cr
        \fi
        \Let@
        \default@tag
        \ifinany@\else\openup\jot\fi
        \column@\z@
        \ialign\bgroup
           &\column@plus
            \hfil
            \strut@
            $\m@th\displaystyle{##}$
           &\column@plus
            $\m@th\displaystyle{{}##}$
            \hfil
            \crcr
 }
 \renewenvironment{aligned}[1][c]{
    \start@aligned{#1}\m@ne
 }{
    \crcr\egroup\egroup
 }
 \makeatother

\title{Maximal Regularity of Parabolic Transmission Problems}
\author{Herbert Amann}
\date{}
\maketitle
\begin{center}
{\small
Dedicated to Matthias Hieber, a~pioneer of maximal regularity,\\
on the occasion of his sixtieth birthday}
\end{center}
\begin{abstract}
Linear reaction-diffusion equations with inhomogeneous boundary and
transmission conditions are shown to possess the property of maximal
\hbox{$L_p$~regularity}. The new feature is the fact that the transmission
interface is allowed to intersect the boundary of the domain transversally.
\end{abstract}
\makeatletter
\def\blfootnote{\xdef\@thefnmark{}\@footnotetext}
\makeatother
\blfootnote{
2010 Mathematics Subject Classification. 35K10 35K57 35K65 58J32\\
Key words and phrases:
Linear reaction-diffusion equations,
inhomogeneous boundary and transmission conditions,
interfaces with boundary intersection,
maximal regularity,
Riemannian manifolds with bounded geometry and singularities,
weighted Sobolev spaces.}

\setcounter{section}{0}
\section{Introduction}\label{sec-I}
The emerging and understanding of the theory of maximal regularity for
par\-a\-bol\-ic differential equations, which took place within the last
three or
so decades, has provided a firm basis for a successful handling of many
challenging nonlinear problems. Among them, phase transition issues play a
particularly prominent role. The impressive progress which has been made in
this field with the help of maximal regularity techniques is
well-documented in the book by J.~Pr{\"u}ss and G.~Simonett~\cite{PrS16a}.
The reader may also consult the extensive list of references
and the `Bibliographic Comments' in~\cite{PrS16a} for works of
other authors and historical developments.

\par
The relevant mathematical setup is usually placed in the framework of
parabolic equations in bounded Euclidean domains, the interface
being modeled as a hypersurface. In most works known to the author it is
assumed that the interface lies in the interior of the domain. Noteworthy
exceptions are the papers by
M.~Wilke~\cite{Wil17a}, J.~Pr\"{u}ss, G.~Simonett, and
M.~Wilke~\cite{PSW19a}, H.~Abels, M.~Rauchecker, and M.~Wilke~\cite{ARW19a},
and M.~Rauchecker~\cite{Rau20a} who study various important parabolic free
boundary problems, presupposing that the membrane makes a ninety degree
boundary contact. In addition, in all of them, except for~\cite{ARW19a},
a~capillary (i.e.,~cylindrical) geometry is being studied.
The same
ninety degree condition is employed by H.~Garcke and
M.~Rauchecker~\cite{GaR19a} who carry out a linearized stability
computation at a stationary solution of a Mullins--Sekerka flow in~a
two-dimensional bounded domain.

\par
The assumption of the ninety degree contact considerably simplifies the
analysis since it allows to use reflection arguments. This does not apply in
the case of general transversal intersection.

\par
The only paper, we are aware of, in which a general contact angle is being
considered is the one by
Ph.~Lauren\c{c}ot and Ch.~Walker~\cite{LaW20a}. These authors establish the
unique solvability in the strong \hbox{$L_2$~sense} of a two-dimensional
stationary transmission problem taking advantage of a particularly
favorable geometric setting.

\par
Elliptic problems with boundary and transmission conditions have also been
investigated in a series of papers by V.~Nistor and coworkers
\cite{BMNZ10a}, \cite{LMN10a}, \cite{LNQ13a}, and~\cite{MaNi10a}. The
motivation for these works stems from the desire to get optimal convergence
rates for approximations used for numerical computations. Although these
authors employ weighted \hbox{$L_2$~Sobolev} spaces, their methods
and results are quite different from the ones presented here.

\par
In this paper we establish the maximal regularity of linear
inhomogeneous parabolic
transmission boundary value problems for the case where the interface
intersects the boundary transversally. This is achieved by allowing the
equations to degenerate near the intersection manifold and working in
suitable weighted Sobolev spaces. We restrict ourselves to the simplest
case of a fixed membrane and a single reaction-diffusion equation.

\par
In a forthcoming publication we shall use our present result to establish the
local well-posedness of quasilinear equations with nonlinear boundary and
transmission conditions.

\par
The author is deeply grateful to G.~Simonett for carefully reading the first
draft of this paper, valuable suggestions, and pointing out misprints,
errors, and the above references to related moving boundary problems.

\section{The Main Result}\label{sec-R}
Now we outline---in a slightly sketchy way---the
main result of this paper. Precise definitions of notions, facts, and
function spaces which we use here without further explanation, are given in
the subsequent sections.

\par
Let $\Om$ be a bounded domain in~$\BR^m$,
\,\hb{m\geq2}, with a smooth boundary~$\Ga$ lying locally on one side
of~$\Om$. By a membrane in~$\oO$ we mean a smooth oriented
hypersurface~$S$ of (the manifold)~$\oO$ with~a (possibly empty)
boundary~$\Sa$ such that
\hb{S\cap\Ga=\Sa}. Thus $S$~lies in~$\Om$ if
\hb{\Sa=\es}. Otherwise, $\Sa$ is an
\hb{(m-2)}-dimensional oriented smooth submanifold of~$\Ga$. In this case
it is assumed that $S$ and~$\Ga$ intersect transversally. Note that
we do not require that $S$~be connected. Hence, even if
\hb{\Sa\neq\es}, there may exist interior membranes. However, the focus
in this paper is on membranes with boundary. Thus we assume until further
notice that
\hb{\Sa\neq\es}.

\par
We denote by~$\nu$ the inner (unit) normal (vector field) on~$\Ga$ and
by~$\nu_S$ the positive normal on~$S$. (Thus
\hb{\nu_S(x)\in T_x\oO=T_x\BR^m=\{x\}\times\BR^m}, the latter being also
identified with
\hb{x+\BR^m\is\BR^m} for
\hb{x\in S}.) As usual,
\hb{\eesdot=\eesdot_S} is the jump across~$S$. We fix any
\hb{T\in(0,\iy)} and set
\hb{J=J_T:=[0,T]}.

\par
Of concern in this paper are linear reaction-diffusion equations with
nonhomogeneous boundary and transmission conditions of the following form.

\par
Set
$$
\bal
\cA u
&:=-\tdiv(a\grad u),\ \cB u:=\ga a\pl_\nu u,\cr
\cC^0 u
&:=\eea{u},\ \cC^1u:=\eea{a\pl_{\nu_S}u},\ \cC=(\cC^0,\cC^1),
\eal
\npbd
$$
with $\ga$~being the trace operator on~$\Ga$. We assume (for the moment)
that
\hb{a\in\bar C^1\bigl((\oO\ssm S)\times J\bigr)} and
\hb{a>0}. A~bar over a symbol for a standard function space means that its
elements may undergo jumps across~$S$. (The usual definitions based on
decompositions of
\hb{\oO\ssm S} in `inner' and `outer' domains cannot be used since
\hb{\oO\ssm S} may be connected.) Then the problem under investigation reads:
\begin{equation}\label{R.RD}
\bal
\pl_tu+\cA u
&=f
 &&\text{ on }(\oO\ssm S)\times J,\cr
\cB u
&=\vp
 &&\text{ on }(\Ga\ssm\Sa)\times J,\cr
\cC u
&=\psi
 &&\text{ on }(S\ssm\Sa)\times J,\cr
\ga_0u
&=u_0
 &&\text{ on }(\oO\ssm S)\times\{0\},\cr
\eal
\npbd
\end{equation}
where $\ga_0$~is the trace operator at
\hb{t=0}.\po

\par
We are interested in the strong \hbox{$L_p$~solvability} of \eqref{R.RD},
that is, in solutions possessing second order space derivatives in~$L_p$.
However,
since $S$~intersects~$\Ga$, we cannot hope to get solutions which possess
this regularity up to~$\Sa$. Instead, it is to be expected that the
derivatives of~$u$ blow up as we approach~$\Sa$. For this reason
we set up our problem in weighted Sobolev spaces where the weights
control the behavior of~$\pa u$ for
\hb{0\leq|\al|\leq2} in relation to the distance from~$\Sa$. This requires
that the differential operator is adapted to such a setting, which means
that the adapted `diffusion coefficient' tends to zero near~$\Sa$.
In other words: we will have to deal with parabolic problems which
degenerate near~$\Sa$. To describe the situation precisely, we introduce
curvilinear coordinates near~$\Sa$ as follows.

\par
Since $\Sa$~is an oriented hypersurface in~$\Ga$, there exists a unique
positive normal vector field~$\mu$ on~$\Sa$ in~$\Ga$. Given
\hb{\sa\in\Sa}, we write~$\mu(\cdot,\sa)$ for the unique geodesic in~$\Ga$
satisfying
\hb{\mu(0,\sa)=\sa} and
\hb{\dot\mu(0,\sa)=\mu(\sa)}. Similarly, for each
\hb{y\in\Ga} we set
\hb{\nu(\xi,y):=y+\xi\nu(y)} for
\hb{\xi\geq0}. Then we can choose
\hb{\ve\in(0,1)} and a neighborhood~$\wt{U}(\ve)$ of~$\Sa$ in~$\oO$
with the following properties: for each
\hb{x\in\wt{U}(\ve)} there exists  a unique triple
$$
(\xi,\eta,\sa)\in N(\ve)\times\Sa
\qa N(\ve):=[0,\ve)\times(-\ve,\ve),
$$
such that
\begin{equation}\label{R.x}
x=x(\xi,\eta,\sa):=\nu\bigl(\xi,\mu(\eta,\sa)\bigr).
\npbd
\end{equation}
Thus
\hb{x\in\Ga\cap\wt{U}(\ve)} iff
\hb{(\xi,\eta,\sa)\in\{0\}\times(-\ve,\ve)\times\Sa}.\po

\par
Now we define curvilinear derivatives for
\hb{u\in C^2\bigl(\wt{U}(\ve)\bigr)} by
\begin{equation}\label{R.nu}
\pl_\nu u(x)=\pl_1(u\circ x)(\xi,\eta,\sa)
\qb \pl_\mu u(x):=\pl_2(u\circ x)(\xi,\eta,\sa)
\npbd
\end{equation}
for
\hb{x\in\wt{U}(\ve)}. It follows that\footnote{If
\hb{m=2}, then the last term must be disregarded. It is understood that
similar interpretations and adaptions are to be made throughout this paper.}
\begin{equation}\label{R.Auv}
\cA u
=-\bigl(\pl_\nu(a\pl_\nu u)+\pl_\mu(a\pl_\mu u)
+\tdiv_\Sa(a\grad_\Sa u)\bigr)
\end{equation}
on~$\wt{U}(\ve)$,
where $\tdiv_\Sa$ and~$\grad_\Sa$ denote the divergence and the gradient,
respectively, in~$\Sa$ (with respect to the Riemannian metric~$g_\Sa$
induced by the one of~$\Ga$ which, in turn, is induced by the
Euclidean metric on~$\oO$).

\par
For~$x$ given by~\eqref{R.x}, we set
\begin{equation}\label{R.r}
r(x):=\sqrt{\xi^2+\eta^2}
\qa (\xi,\eta)\in N(\ve),
\end{equation}
which is the geodesic distance in~$\oO$ from~$x$ to~$\Sa$
(and not, in general, the distance in the ambient space~$\BR^m$). We fix
\hb{\om\in C^\iy\bigl(N(\ve),[0,1]\bigr)}, depending only on~$r$, such that
\hb{\om\sn N(\ve/3)=1} and
\hb{\supp(\om)\is N(2\ve/3)} and set
\begin{equation}\label{R.rh}
\rho:=1-\om+r\om.
\end{equation}
Then we define on
\begin{equation}\label{R.U}
U:=U(\ve):=\wt{U}(\ve)\ssm\Sa
\end{equation}
a~singular linear reaction-diffusion operator~$\cA_U$ by
\begin{equation}\label{R.AU}
\cA_Uu:=-\rho^2\bigl(\pl_\nu(a\pl_\nu u)+\pl_\mu(a\pl_\mu u)\bigr)
-\tdiv_\Sa(a\grad_\Sa u)
\end{equation}
for
\hb{u\in\bar C^2(U\ssm S)}. The corresponding singular boundary
operator is given by
\begin{equation}\label{R.BU}
\cB_Uu:=\ga a\rho\pl_\nu u.
\end{equation}

\par
Since $S$~intersects~$\Ga$ transversally, it follows that there exists
a smooth function
\hb{s\sco[0,\ve)\times\Sa\ra(-\ve,\ve)} such that
\hb{s(0,\sa)=0} for
\hb{\sa\in\Sa} and
\begin{equation}\label{R.xS}
x\in\wt{U}(\ve)\cap S
\quad\text{iff}\quad
x=\bigl(\xi,s(\xi,\sa),\sa\bigr)
\qa (\xi,\sa)\in[0,\ve)\times\Sa.
\end{equation}
Using this we associate  with~$\cA_U$ a~transmission
operator~$\cC_U$ on~$U$ by setting
$$
\bal
\cC_U^0u
&:=\eea{u}_{U\cap S},\cr
\cC_U^1u
&:=\beea{a\bigl(\nu_S^1\pl_\nu u+\nu_S^2\pl_\mu u
+\nu_S^3(\grad_\Sa u\sn\grad_\Sa s)\bigr)_\Sa}_{U\cap S}
\eal
$$
for
\hb{u\in\bar C^2(U\ssm S)}, where
\hb{\prsn_\Sa=g_\Sa} and
$$
(\nu_S^1,\nu_S^2,\nu_S^3)
:=(\pl_\nu s,-1,1)\Big/\sqrt{1+(\pl_\nu s)^2+|\grad_\Sa s|_\Sa^2}.
$$
Now we define a~\emph{singular transmission boundary value problem on}
\hb{\oO\ssm S}~by putting
\hb{V:=\oO\,\big\backslash\wt{U}(2\ve/3)} and
$$
(\cA_r,\cB_r,\cC_r):=
\left\{
\bal
{}
&(\cA,\cB,\cC)
 &&\ \text{on }V,\cr
&(\cA_U,\cB_U,\cC_U)
 &&\ \text{on }U.
\eal
\right.
\npbd
$$
It follows from \eqref{R.Auv} and the properties of~$\rho$ that this
definition is unambiguous.\po

\par
To introduce weighted Sobolev spaces on
\hb{U\ssm S} we put
\begin{equation}\label{R.us}
\bal
{}
\dl u\dr^2:=|u|^2
&+|r\pl_\nu u|^2
 +|r\pl_\mu u|^2\cr
&+|(r\pl_\nu)^2u|^2
 +|r\pl_\nu(r\pl_\mu u)|^2
 +|r\pl_\mu(r\pl_\nu u)|^2
 +|(r\pl_\mu)^2u|^2\cr
&+|\naSa u|^2+|\naSa^2u|^2,
\eal
\end{equation}
where $\naSa$~is the Levi--Civita connection on~$\Sa$
for the metric~$g_\Sa$. Moreover,
\hb{1<p<\iy} and
\begin{equation}\label{R.WU}
\|u\|_{\bar W_{\coW p}^2(U\setminus S;r)}
:=\Bigl(\int_{U\setminus S}\dl u\dr^p\,\frac{d(\xi,\eta)}{r^2}
\,d\vol_\Sa\Bigr)^{1/p}.
\end{equation}
Then
\hb{\bar W_{\coW p}^2(U\ssm S;r)} is the completion of
\hb{\bar C^2(U\ssm S)} in
\hb{L_{1,\loc}(U\ssm S)} with respect to the norm~
\hb{\Vsdot_{\bar W_{\coW p}^2(U\setminus S;r)}}.

\par
The (global) weighted Sobolev space
$$
\cX_p^2:=\bar W_{\coW p}^2(\oO\ssm S;r)
$$
consists of all
\hb{u\in L_{1,\loc}(\oO\ssm S)} with
\hb{u\bsn U\in\bar W_{\coW p}^2(U\ssm S;r)} and
\hb{u\bsn V\in\bar W_{\coW p}^2(V\ssm S)}. It is a Banach space
with the norm
$$
u\mt\big\|u\sn U\big\|_{\bar W_{\coW p}^2(U\setminus S;r)}
+\big\|u\sn V\big\|_{\bar W_{\coW p}^2(V\setminus S)},
$$
whose topology is independent of the specific choice of $\ve$ and~$\om$.
Similarly, the Lebesgue space
$$
\cX_p^0:=\bar W_{\coW p}^0(\Om\ssm S;r)
$$
is obtained by replacing~$\dl u\dr$ in \eqref{R.WU} by~$|u|$. Moreover,
\begin{equation}\label{R.int}
\cX_p^{2-2/p}:=
\bar W_{\coW p}^{2-2/p}(\oO\ssm S;r)
:=(\cX_p^0,\cX_p^2)_{1-1/p,p},
\npbd
\end{equation}
where
\hb{\prsn_{\ta,p}} is the real interpolation functor of exponent~$\ta$.\po

\par
We also need time-dependent anisotropic spaces. For this we use the notation
\hb{s/\mf2:=(s,\,s/2)},
\ \hb{0\leq s\leq2}. Then
$$
\cX_p^{2/\mf2}
:=\bar W_{\coW p}^{2/\mf2}\bigl((\oO\ssm S)\times J;r\bigr)
:=L_p(J,\cX_p^2)\cap W_{\coW p}^1(J,\cX_p^0)
$$
and
\hb{\cX_p^{0/\mf2}:=L_p(J,\cX_p^0)}. If
\hb{X\in\{\Ga,S\}} and
\hb{s\in\{1-1/p,\ 2-1/p\}}, then
$$
\bar W_{\coW p}^{s/\mf2}\bigl((X\ssm\Sa)\times J;r\bigr)
:=L_p\bigl(J,\bar W_{\coW p}^s(X\ssm\Sa;r)\bigr)
\cap W_{\coW p}^{s/2}\bigl(J,L_p(X\ssm\Sa;r)\bigr).
$$
Here the
\hb{\bar W_{\coW p}^s(X\ssm\Sa;r)} are trace spaces of~$\cX_p^2$
(cf.~\eqref{MB.Ga} and \eqref{MB.S}). Moreover,
$$
\bal
\cY_p
 :=\bar W_{\coW p}^{(1-1/p)/\mf2}\bigl((\Ga\ssm\Sa)\times J;r\bigr)
&\oplus\bar W_{\coW p}^{(2-1/p)/\mf2}\bigl((S\ssm\Sa)\times J;r\bigr)\cr
&\oplus\bar W_{\coW p}^{(1-1/p)/\mf2}\bigl((S\ssm\Sa)\times J;r\bigr).
\eal
$$

\par
By
\hb{\bar{BC}(\oO\ssm S)} we mean the space of bounded and continuous
functions (with possible jumps across~$S$), endowed with the maximum norm.
Then
\hb{\bar{BC}^1(\oO\ssm S;r)} is the Banach space of all
\hb{u\in\bar{BC}(\oO\ssm S)} with
\hb{\pl_ju\in\bar{BC}(V\ssm S)},
\ \hb{1\leq j\leq m}, and
$$
\rho\pl_\nu u,\rho\pl_\mu u\in\bar{BC}(U\ssm S)
\qb u\sn\Sa\in BC^1(\Sa).
$$
Furthermore,
$$
\bar{BC}^{1/\mf2}\bigl((\oO\ssm S)\times J;r\bigr)
:=C\bigl(J,\bar{BC}^1(\oO\ssm S;r)\bigr)
\cap C^{1/2}\bigl(J,\bar{BC}(\oO\ssm S)\bigr).
$$
To indicate the nonautonomous structure of \eqref{R.RD}, we write
\hb{a(t):=a(\cdot,t)} and, correspondingly, $\cA(t)$, $\cB(t)$, and~$\cC(t)$.

\par
Now we are ready to formulate the main result of this paper, the optimal
solvability of linear reaction-diffusion transmission
boundary value problems.

\begin{theorem}\label{thm-R.RD}
Let
\hb{1<p<\iy} with
\hb{p\notin\{3/2,\,3\}} and
$$
a\in\bar{BC}^{1/\mf2}\bigl((\oO\ssm S)\times J;r\bigr)
\qa a\geq\ul{\al},
$$
for some
\hb{\ul{\al}\in(0,1)}. Suppose
$$
\bigl(f,(\vp,\psi^0,\psi^1),u_0\bigr)
\in\cX_p^{0/\mf2}\oplus\cY_p\oplus\cX_p^{2-2/p}
$$
and that the following compatibility conditions are satisfied:
$$
\bal
{\rm(i)}\quad
\cC_r^0(0)u_0
&=\psi^0(0),
&\quad\text{if\/ }&&3/2<p<3,\cr
{\rm(ii)}\quad
\cB_r(0)u_0
&=\vp(0),
 \ \cC_r(0)u_0=\psi(0),
&\quad\text{if\/ }&&p>3,
\eal
$$
where
\hb{\psi:=(\psi^0,\psi^1)}. Then
\begin{equation}\label{R.P}
\bal
\bal
\pl_tu+\cA_r(t)u    &=f      &&\text{ on }(\oO\ssm S)\times J,\cr
\cB_r(t)u           &=\vp    &&\text{ on }(\Ga\ssm\Sa)\times J,\cr
\cC_r(t)u           &=\psi   &&\text{ on }(S\times\Sa)\times J,\cr
\ga_0u              &=u_0    &&\text{ on }(\oO\ssm S)\times\{0\}
\eal
\eal
\npbd
\end{equation}
has a unique solution
\hb{u\in\cX_p^{2/\mf2}}. It depends continuously on the data.\po
\end{theorem}

\setcounter{corollary}{1}
\begin{corollary}\label{cor-R.RD}
Suppose $a$~is independent of~$t$, that is,
\hb{a\in\bar{BC}^1(\oO\ssm S;r)}. Set
$$
\cX_{p,0}^2:=
\left\{
\bal
{}
&\ \cX_p^2,
 &\quad 1  &<p<3/2,\cr
&\{\,u\in\cX_p^2\ ;\ \cC^0u=0\,\},
 &\quad 3/2&<p<3,\cr
&\{\,u\in\cX_p^2\ ;\ \cB u=0,\ \cC u=0\,\},
 &\quad 3  &<p<\iy,
\eal
\right.
$$
and
\hb{A_r:=\cA_r\sn\cX_{p,0}^2}. Then~$-A_r$, considered as a
linear operator in~$\cX_p^0$ with domain~$\cX_{p,0}^2$, generates
on~$\cX_p^0$ a strongly continuous analytic semigroup.
\end{corollary}

\begin{proof}
The theorem implies that $A_r$~has the property of maximal
\hbox{$\cX_p^0$~regularity}. This fact is well-known to imply the claim
(e.g.,~\cite[Capter~III]{Ama95a} or~\cite{DHP07a}).
\end{proof}
Theorem~\ref{thm-R.RD} is a special case of the much more general
Theorems \ref{thm-L.MR} and \ref{thm-MB.P}. They also include Dirichlet
boundary conditions and apply to transmission problems in general
Riemannian manifolds with boundary and bounded geometry.

\par
The situation is considerably simpler if
\hb{\Sa=\es}, that is, if only interior transmission hypersurfaces
are present. Of course, if
\hb{S=\es}, then \eqref{R.P} reduces to a linear reaction-diffusion
equation with inhomogeneous boundary conditions. In these cases no
degenerations do occur.

\par
We refrain from considering operators~$\ArBr$ with lower
order terms. This case will be covered by the forthcoming
quasilinear result.

\par
In the case of an interior transmission surface (that is,
\hb{\Sa=\es}) and if $a$~is independent of~$t$, Theorem~\ref{thm-R.RD}
is a special case of Theorem~6.5.1 in~\cite{PrS16a}. The latter
theorem applies to systems and provides an
\hbox{$L_p$-$L_q$} theory.

\par
If
\hb{\Sa\neq\es}, then the basic difficulty in proving Theorem~\ref{thm-R.RD}
stems from the fact that
\hb{\oO\ssm\Sa} and, consequently,
\hb{S\ssm\Sa} and
\hb{\Ga\ssm\Sa}, are no longer compact. The fundamental observation which
makes the proofs work is the fact that we can consider
\hb{\oO\ssm\Sa} as a (noncompact) Riemannian manifold with a metric~$g$
which coincides on
\hb{U(\ve/3)} with the singular metric
\hb{r^{-2}d\nu\otimes d\mu+g_\Sa} and on~$V$ with the Euclidean metric.
With respect to this metric, $\cA_r$~is a uniformly elliptic operator.

\par
Theorems \ref{thm-H.C}  and~\ref{thm-S.M} show that
\hb{(\oO\ssm\Sa,\,g)} is a uniformly regular Riemannian manifold in the sense
of~\cite{Ama12b}. Thus we are led to consider linear parabolic equations with
boundary and transmission conditions on such manifolds. As in the compact
case, by means of local coordinates the problem is reduced to Euclidean
settings. However, since we have to deal with noncompact manifolds, we have
to handle simultaneously infinitely many model problems. In order for this
technique to work, we have to establish uniform estimates which are in a
suitable sense independent of the specific local coordinates. In addition,
special care has to be taken in `gluing together the local model problems'.
These are no points to worry about in the compact case.

\par
In our earlier paper~\cite{Ama17a} we have established an optimal existence
theory for linear parabolic equations on uniformly regular Riemannian
manifolds without boundary. The present proof extends those arguments to
the case of manifolds with boundary. The presence of boundary and
transmission conditions adds considerably to the complexity of the problem
and makes the paper rather heavy.

\par
In Section~\ref{sec-U} we collect the needed background information.
In the subsequent two sections we establish the differential geometric
foundation of transmission surfaces in uniformly regular and singular
Riemannian manifolds.

\par
After having introduced the relevant function spaces in
Section~\ref{sec-F}, we present in Section~\ref{sec-L} the basic maximal
regularity theorem in anisotropic Sobolev spaces for linear
non-autonomous reaction-diffusion equations with nonhomogeneous boundary and
transmission conditions on general uniformly regular Riemannian manifolds.
Its rather complex proof occupies the next
five sections. Finally, in the last section it is shown that our general
results apply to the Euclidean setting presented here.

\section{Uniformly Regular Riemannian Manifolds}\label{sec-U}
In this section we recall the definition of uniformly regular Riemannian
manifolds and collect those
properties of which we will make use. Details can be found in
\cite{Ama12c}, \cite{Ama12b}, \cite{Ama15a}, and in the comprehensive
presentation~\cite{AmaVolIII21a}. Thus we shall be rather brief.

\par
We use standard notation from differential geometry and function space
theory. In particular, an upper, resp.~lower, asterisk on a symbol for a
diffeomorphism denominates the corresponding pull-back, resp.\ push-forward
(of tensors).
By~$c$, resp.~$c(\al)$ etc., we denote constants~
\hb{\geq1} which can vary from occurrence  to occurrence.
Assume $S$~is a nonempty set. On the cone of nonnegative functions on~$S$
we define an equivalence relation~
\hb{{}\sim{}} by
\hb{f\sim g} iff
\hb{f(s)/c\leq g(s)\leq cf(s)},
\ \hb{s\in S}.

\par
An \hbox{$m$-dimensional} manifold is a separable metrizable space equipped
with an \hbox{$m$-dimensional} smooth structure. We always work in the
smooth category.

\par
Let $M$ be an \hbox{$m$-dimensional} manifold with or without boundary. If
$\ka$~is a local chart, then we use~$U_{\coU\ka}$ for its domain, the
coordinate patch associated with~$\ka$. The chart is normalized if
\hb{\ka(U_{\coU\ka})=Q_\ka^m}, where
\hb{Q_\ka^m=(-1,1)^m} if
\hb{U\is\ci M}, the interior of~$M$, and
\hb{Q_\ka^m=[0,1)\times(-1,1)^{m-1}} otherwise. An atlas~$\gK$ is normalized
if it consists of normalized charts. It is shrinkable if it normalized and
there exists
\hb{r\in(0,1)} such that
\hb{\bigl\{\,\ka^{-1}(rQ_\ka^m)\ ;\ \ka\in\gK\,\bigr\}} is a covering
of~$M$. It has finite multiplicity if there exists
\hb{k\in\BN} such that any intersection of more than~$k$ coordinate
patches is empty.

\bigskip
The atlas~$\gK$ is \emph{uniformly regular}~(ur) if
\begin{equation}\label{U.K}
\bal
{\rm(i)}\quad
&\text{it is shrinkable and has finite multiplicity;}\cr
{\rm(ii)}\quad
&\wk\circ\ka^{-1}\in\BUC^\iy\bigl(\ka(U_\kwk),\BR^m\bigr)
 \text{ and }\|\wk\circ\ka^{-1}\|_{k,\iy}\leq c(k)\cr
\noalign{\vskip-1\jot}
&\text{for }\ka,\wk\in\gK\text{ and }k\in\BN,
 \text{ where }U_\kwk:=U_{\coU\ka}\cap U_\wk.
\eal
\end{equation}
Two ur atlases $\gK$ and~$\wt{\gK}$ are equivalent if
\begin{equation}\label{U.e}
\kern-8pt
\bal
{\rm(i)}\quad
&\text{there exists $k\in\BN$ such that each coordinate patch of $\gK$}\cr
\noalign{\vskip-1.5\jot}
&\text{meets at most $k$ coordinate patches of $\wt{\gK}$,
 and vice versa;}\cr
{\rm(ii)}\quad
&\text{condition~\eqref{U.K}(ii) holds for all }
 (\ka,\wt{\ka})\text{ and }(\wt{\ka},\ka)\text{ belonging to }
 \gK\times\wt{\gK}.
\eal
\kern-10pt
\end{equation}
This defines an equivalence relation in the class of all ur~atlases. An
equivalence class thereof is a ur~structure. By~a \emph{ur~manifold}
we mean a manifold equipped with a ur~structure.
A~Riemannian metric~$g$ on a~ur manifold~$M$ is~\emph{ur} if,
given a~ur atlas~$\gK$,
\begin{equation}\label{U.g}
\bal
{\rm(i)}\quad
&\ka_*g\sim g_m,\ \ka\in\gK;\cr
{\rm(ii)}\quad
&\|\ka_*g\|_{k,\iy}\leq c(k),\ \ka\in\gK,\ k\in\BN.
\eal
\end{equation}
Here
\hb{g_m:=\prsn=dx^2} is the Euclidean
metric\footnote{We use the same symbol for a Riemannian metric
and its restrictions to sub\-manifolds of the same dimension.}
on~$\BR^m$ and (i)~is understood
in the sense of quadratic forms. This concept is well-defined,
independently of the specific~$\gK$. A~\emph{uniformly regular Riemannian}
(urR) \emph{manifold} is~a ur~manifold, endowed with~a urR metric.

\setcounter{remarks}{0}
\begin{remarks}\label{rem-U.R}
(a)
Given a (nonempty)
subset~$S$ of~$M$ and an atlas~$\gK$,
$$
\gK_S:=\{\,\ka\in\gK\ ;\ U_{\coU\ka}\cap S\neq\es\,\}\ .
$$
We say that $\gK$~is normalized \emph{on}~$S$, resp.\ has finite
multiplicity \emph{on}~$S$, resp.\ is shrinkable \emph{on}~$S$
if $\gK_S$~possesses the respective properties. Moreover, $\gK$~is~ur
\emph{on}~$S$ if \eqref{U.K} applies with~$\gK$ replaced
by~$\gK_S$. Similarly, two atlases $\gK$ and~$\wt{\gK}$, which are
ur \emph{on}~$S$, are equivalent \emph{on}~$S$ if \eqref{U.e} holds with
$\gK$ and~$\wt{\gK}$ replaced by $\gK_S$  and~$\wt{\gK}_S$, respectively.
This induces a \hbox{ur~structure} \emph{on}~$S$. Finally, $M$~is~ur
\emph{on}~$S$ if it is equipped with~a \hbox{ur~structure} on~$S$.

\par
(b)
Suppose $\gK$~is a ur atlas for~$M$ on~$S$. Given any
\hb{\ve>0}, there exists a ur atlas~$\gK'$ on~$S$ such that
\hb{\diam_g(U_{\coU\ka})<\ve} for
\hb{\ka\in\gK'}, where $\diam_g$~is the diameter with respect to the
Riemannian distance~$d_g$.\qed
\end{remarks}
In the following examples we use the natural ur~structure
(e.g.,~the product ur~structure in Example~\ref{exa-U.ex}(c))
if nothing else is mentioned.

\setcounter{examples}{1}
\begin{examples}\label{exa-U.ex}
(a)
Each compact Riemannian manifold is~a urR manifold and its ur~structure is
unique.

\par
(b)
Let $\Om$ be a bounded domain in~$\BR^m$ with a smooth boundary such that
$\Om$~lies locally on one side of it. Then $(\oO,g_m)$ is~a
urR manifold.

\par
More generally, suppose that $\Om$~is an unbounded open subset of~$\BR^m$
whose boundary is ur in the sense of F.E.~Browder~\cite{Bro59a} (also see
\cite[IV.\S4]{LSU68a}). Then $\oOgm$~is a urR manifold.
In particular, $\Rmgm$ and~$\Hmgm$ are urR manifolds, where
\hb{\BH^m:=\BR_+\times\BR^{m-1}} is the closed right half-space in~$\BR^m$.

\par
(c)
If $\Migi$,
\ \hb{i=1,2}, are urR manifolds and at most one of them has a nonempty
boundary, then
\hb{(M_1\times M_2,\ g_1+g_2)} is~a urR manifold.

(d)
Assume $M$~is a manifold and $N$~a topological space. Let
\hb{f\colon N\ra M} be a homeomorphism. If $\gK$ is an atlas for~$M$, then
\hb{f^*\gK:=\{\,f^*\ka\ ;\ \ka\in\gK\,\}} is an atlas for~$N$ which induces
the smooth `pull-back' structure on~$N$. If $\gK$~is ur,
then $f^*\gK$ also is~ur.

\par
Let $\Mg$ be~a urR manifold. Then
\hb{f^*\Mg:=(N,f^*g)} is~a urR manifold and the map
\hb{f\colon(N,f^*g)\ra\Mg} is an isometric diffeomorphism.\qed
\end{examples}
It follows from these examples, for instance, that the cylinders
\hb{\BR\times M_1} or
\hb{\BR_+\times M_2}, where $M_i$~are compact Riemannian manifolds with
\hb{\pl M_2=\es}, are
urR manifolds. More generally, Riemannian manifolds with cylindrical
ends are urR manifolds (see~\cite{Ama15a} where more examples
are discussed).

\par
Without going into detail, we mention that a Riemannian manifold without
boundary is~a urR manifold iff it has bounded geometry (see~\cite{Ama12b}
for one half of this assertion and \cite{DSS16a} for the other half). Thus,
for example, $(\ci\BH^m,g_m)$~is \emph{not} a urR manifold.
A~Riemannian manifold with boundary is~a urR manifold iff it has bounded
geometry in the sense of Th.~Schick~\cite{Schi01a} (also see
\cite{AGN19a}, \cite{AGN19b}, \cite{AGN19c}, \cite{GSchn13a} for related
definitions). Detailed proofs of these equivalences can be found
in~\cite{AmaVolIII21a}.

\section{Uniformly Regular Hypersurfaces}\label{sec-H}
Let $\Mg$ be an oriented Riemannian manifold with (possibly empty)
boundary~$\Ga$. If it is not empty, then there exists
a unique \emph{inner} (unit) \emph{normal} vector field
\hb{\nu=\nu_\Ga} on~$\Ga$, that is, a~smooth section of~$T_\Ga M$,
the restriction
of the tangent bundle $TM$ of~$M$ to~$\Ga$. Furthermore, $\Ga$~is oriented
by the inner normal in the usual sense.

\par
Suppose that $S$~is an oriented hypersurface in~$\ci M$, an
embedded submanifold of codimension~$1$. Then there is a unique
positive (unit) normal vector field~$\nu_S$ on~$S$,
where `positive' means that
\hb{\bigl[\nu_S(p),\ba_1,\ldots,\ba_{m-1}\bigr]} is a positive basis
for~$T_pM$ if
\hb{[\ba_1,\ldots,\ba_{m-1}]} is one for~$T_pS$.

\par
Let
\hb{Z\in\{\Ga,S\}}. Then we write
$$
\ga_p^Z(t):=\exp_p\bigl(t\nu_Z(p)\bigr)
\qa t\in I_Z\bigl(\ve(p)\bigr).
$$
This means that, given
\hb{p\in Z},
\ $\ga_p^Z$~is the unique geodesic in~$M$ satisfying
\hb{\ga_p^Z(0)=p} and
\hb{\dot\ga_p^Z(0)=\nu_Z(p)} and being defined (at least)
on~$I_Z\bigl(\ve(p)\bigr)$, where
\hb{I_\Ga\bigl(\ve(p)\bigr)=\bigl[0,\ve(p)\bigr)} and
\hb{I_S\bigl(\ve(p)\bigr)=\bigl(-\ve(p),\ve(p)\bigr)} for some
\hb{\ve(p)>0}. Note that
\hb{\ga_p^\Ga(t)\in\ci M} for
\hb{t>0}.

\par
We say that $Z$~has a \emph{uniform} normal geodesic \emph{tubular
neighborhood} of width~$\ve$ if the following is true: there exist
\hb{\ve>0} and an open neighborhood~$Z(\ve)$ of~$Z$ in~$M$
such that
\begin{equation}\label{H.pN}
\vp_Z\sco Z(\ve)\ra I_Z(\ve)\times Z
\quad\text{with}\quad
\vp_Z^{-1}(t,p)=\ga_p^Z(t)
\end{equation}
is a diffeomorphism satisfying
\hb{\vp_Z(Z)=\{0\}\times Z}. If
\hb{Z=\Ga}, then a uniform tubular neighborhood is~a uniform \emph{collar}.

\par
Given any embedded submanifold~$C$ of~$M$, with or without boundary,
we denote by~$g_C$ the pull-back metric~$\ia^*g$, where
\hb{\ia\sco C\hr M} is the natural embedding.

\par
Now we suppose that
\begin{equation}\label{H.M}
\Mg\text{ is an $m$-dimensional oriented urR manifold}.
\npbd
\end{equation}
This means that there exists an oriented ur atlas for~$M$.\po

\par
Let $S$ be a hypersurface with boundary~$\Sa$ such that
\hb{\Sa=S\cap\Ga}. Thus
\hb{S\is\ci M} if
\hb{\Sa=\es}. An atlas~$\gK$ for~$M$ is \hbox{$S$\emph{-adapted}}
if for each
\hb{\ka\in\gK_S} one of the following alternatives applies:
$$
\bal
{\rm(i)}\quad
&\ka\notin\gK_\Ga.\text{ Then }Q_\ka^m=(-1,1)^m\text{ and}\cr
&\ka(S\cap U_{\coU\ka})=\{0\}\times(-1,1)^{m-1};\cr
{\rm(ii)}\quad
&\ka\in\gK_\Ga.
 \text{ Then }Q_\ka^m=[0,1)\times(-1,1)^{m-1},\cr
&\ka(\Ga\cap U_{\coU\ka})=\{0\}\times(-1,1)^{m-1},\text{ and}\cr
&\ka(\Sa\cap U_{\coU\ka})=\{0\}^2\times(-1,1)^{m-2}.
\eal
$$
Then $S$~is~a \emph{regularly embedded hypersurface in}~$M$,
a~\emph{membrane} for short, if there exists an oriented ur atlas~$\gK$
for~$M$ which is \hbox{$S$-adapted}.

\par
Let $S$ be a membrane. Each \hbox{$S$-adapted} atlas for~$M$ induces
(by restriction) a~ur structure and~a (natural) orientation on~$S$.
Moreover, the ur structure and the orientation of~$S$ are independent
of the specific choice of~$\gK$.

\par
For the proof of all this and the following theorem we refer
to~\cite{AmaVolIII21a}.

\setcounter{theorem}{0}
\begin{theorem}\label{thm-H.S}
Let \eqref{H.M} be satisfied and suppose $S$~is a membrane in~$M$. Assume
\hb{Z\in\{\Ga,S\}}. Then
\begin{itemize}
\item[{\rm(i)}]
$(Z,g_Z)$~is an
(\hb{m-1})-dimensional oriented urR manifold.
\item[{\rm(ii)}]
If
\hb{\Sa=\pl S\neq\es}, then $\Sa$~is a membrane in~$\Ga$ without boundary.
\item[{\rm(iii)}]
Let
\hb{\Sa=\es} if
\hb{Z=S}. Then $Z$~has a uniform tubular neighborhood
$$
\vp_Z\sco Z(\ve)\ra I_Z(\ve)\times Z
$$
and
\hb{\vp_{Z*}g\sim ds^2+g_Z}.
Moreover, $\vp_Z$~is an orientation preserving diffeomorphism.
\item[{\rm(iv)}]
Suppose
\hb{\Sa\neq\es}. Then, given
\hb{\rho>0}, there exists
\hb{\ve(\rho)>0} such that
$$
S\cap\bigl\{\,q\in M\ ;\ d_g(q,\Ga)>\rho\,\bigr\}
\npbd
$$
has a uniform tubular neighborhood of width~$\ve(\rho)$ in~$\ci M$.\po
\end{itemize}
\end{theorem}
Now we suppose that $S$~is a membrane with
\hb{\Sa\neq\es}. It follows from (ii) and~(iii) that
$\Sa$~has a uniform tubular neighborhood
\hb{\psi\sco\Sa^\Ga(\ve)\ra(-\ve,\ve)\times\Sa} \emph{in}~$\Ga$ for some
\hb{\ve>0}. By part~(iii), $\Ga$~has a uniform collar
\hb{\vp\sco\Ga(\ve)\ra[0,\ve)\times\Ga} in~$M$, where we assume without
loss of generality that $\vp$ and~$\psi$ are of the same width. Then
$$
\Sa(\ve):=\bigl\{\,\ga_q^\Ga(t)\ ;\ q\in\Sa^\Ga(\ve),\ 0\leq t<\ve\,\bigr\}
$$
is an open neighborhood of~$\Sa$ in~$M$ and
$$
\chi\sco\Sa(\ve)\ra[0,\ve)\times(-\ve,\ve)\times\Sa
\quad\text{with}\quad
\chi^{-1}(x,y,\sa):=\vp^{-1}\bigl(x,\psi^{-1}(y,\sa)\bigr)
$$
is an orientation preserving diffeomorphism, a~\emph{tubular
neighborhood of~$\Sa$ in~$M$} of width~$\ve$.

\par
We refer once more to~\cite{AmaVolIII21a} for the proof of the next
theorem. Henceforth,
\hb{h:=g_\Sa} and
\hb{N(\ve):=[0,\ve)\times(-\ve,\ve)}.

\begin{theorem}\label{thm-H.Si}
Assume \eqref{H.M} and $S$~is a membrane with nonempty
boundary~$\Sa$. Then
\begin{equation}\label{H.g}
\chi_*g\sim dx^2+dy^2+h.
\end{equation}
\end{theorem}
It follows that
$$
\Sa_\sa(\ve):=\chi^{-1}\bigl(N(\ve)\times\{\sa\}\bigr)
$$
is for each
\hb{\sa\in\Sa} a~\hbox{$2$-dimensional} submanifold of~$\Sa(\ve)$ and
\hb{S\cap\Sa_\sa(\ve)} is~a \hbox{$1$-dimensional} submanifold
of~$\Sa_\sa(\ve)$.

\setcounter{remark}{2}
\begin{remark}\label{rem-H.2}
\uti{The two-dimensional case}
Suppose
\hb{\dim(M)=2} and
\hb{\Sa\neq\es}. It follows from Theorem~\ref{thm-H.S}(ii) and the fact that
$M$~has a countable base that $\Sa$~is a countable discrete subspace of~$M$.
Thus we can find
\hb{\ve>0} with the following properties: if we denote by
\hb{\psi^{-1}(\cdot,\sa)\sco(-\ve,\ve)\ra\Ga} the local arc-length
parametrization of~$\Ga$ with
\hb{\psi^{-1}(0,\sa)=\sa} for
\hb{\sa\in\Sa}, then the above definition of~$\chi$ applies and defines
a~tubular neighborhood of~$\Sa$ in~$M$ of width~$\ve$.

\par
Note that $\Sa(\ve)$~is the countable pair-wise disjoint union
of~$\Sa_\sa(\ve)$,
\ \hb{\sa\in\Sa}. The term~$+h$ in \eqref{H.g} (and everywhere else) has to
be disregarded and the volume measure of~$\Sa$ is the counting measure.
Thus in this case integration with respect to~$d\vol_\Sa$ reduces to
summation over
\hb{\sa\in\Sa}.\qed
\end{remark}
Now we restrict the class of membranes under consideration
by requiring that $S$~\emph{intersects~$\Ga$ uniformly transversally}.
This means the following: there exists
\hb{f\in C^\iy\bigl([0,\ve)\times\Sa,\,(-\ve,\ve)\bigr)} such that, setting
\hb{f_\sa:=f(\cdot,\sa)},
\begin{equation}\label{H.f}
\bal
{\rm(i)}\quad
&f_\sa(0)=0,\ \sa\in\Sa;\cr
{\rm(ii)}\quad
&\text{Given $\ol{\ve}\in(0,\ve)$,
there exists $\ol{\rho}\in(0,\ve)$ with}\cr
&\qquad\quad
 |f_\sa(x)|\leq\ol{\rho},\ 0\leq x\leq\ol{\ve},\ \sa\in\Sa;\cr
 {\rm(iii)}\quad
&|\pl f_\sa(0)|\leq c,\ \sa\in\Sa;\cr
 {\rm(iv)}\quad
&\chi\bigl(S\cap\Sa_\sa(\ve)\bigr)=\graph(f_\sa)\times\{\sa\},\ \sa\in\Sa.
\eal
\end{equation}
In general, a~submanifold~$C$ of a manifold~$B$ intersects~$\pl B$
transversally if
$$
T_pC+T_p\pl B=T_pB
\qa p\in\pl B.
$$

\par
The following theorem furnishes an important large class of urR
manifolds and membranes intersecting the boundary uniformly transversally.

\setcounter{theorem}{3}
\begin{theorem}\label{thm-H.C} Let $\Mg$ be a compact oriented Riemannian
manifold with boundary~$\Ga$.
Assume $S$~is an oriented hypersurface in~$M$ with nonempty boundary
\hb{\Sa\is\Ga} and $S$~intersects~$\Ga$ transversally. Then $\Mg$~is a urR
manifold and $S$~is a membrane intersecting~$\Ga$ uniformly transversally.
\end{theorem}

\begin{proof}
Example~\ref{exa-U.ex}(a) guarantees that $\Mg$~is an oriented urR
manifold. Thus $(\Ga,g_\Ga)$ is an oriented urR manifold by
Theorem~\ref{thm-H.S}(i). Since $S$~intersects~$\Ga$ transversally, it is a
well-known consequence of the implicit function theorem that $\Sa$~is a
compact hypersurface in~$\Ga$ without boundary. It is oriented, being the
boundary of the oriented manifold~$S$. Hence, invoking
Example~\ref{exa-U.ex}(a) once more, $(\Sa,g_\Sa)$ is an oriented urR
manifold. As it is compact, it has a uniform tubular neighborhood
in~$\Ga$. Thus, $\Ga$~having a uniform collar, $\Sa$~has a uniform
tubular neighborhood~$\chi$ in~$M$ of some width~$\ve$.

\par
Since $S$~intersects~$\Ga$ transversally,
\hb{\chi\bigl(S\cap\Sa_\sa(\ve)\bigr)} can be represented as the graph
of a smooth function
\hb{f_\sa\sco[0,\ve)\ra(-\ve,\ve)} with
\hb{f_\sa(0)=0}, and $f_\sa$~depends smoothly on
\hb{\sa\in\Sa}. The compactness of~$\Sa$ implies that \eqref{H.f} is true.
Hence $S$~intersects~$\Ga$ uniformly transversally.
Now, due to the compactness of~$S$, it is not difficult to see that
$S$~is a regularly embedded submanifold of~$M$. The theorem is proved.
\end{proof}

\setcounter{remarks}{4}
\begin{remarks}\label{rem-H.C}
(a)
This theorem applies to the case
\hb{\Mg=\oOgm} considered in Section~\ref{sec-R}.

\par
(b)
Suppose $\Mg$~is an oriented urR manifold and $S$~a membrane without
boundary. Then the fact that $S$~has a uniform tubular neighborhood
in~$\ci M$ prevents~$S$ from either
reaching~$\Ga$ or `collapsing' at infinity.\qed
\end{remarks}

\section{The Singular Manifold}\label{sec-S}
In this section
\begin{equation}\label{S.ass}
\bal
{}
&\Mg\text{ is an oriented urR manifold and}\cr
\noalign{\vskip-1\jot}
&S\text{ a membrane with nonempty boundary }\Sa\cr
\noalign{\vskip-1\jot}&\text{such that $S$ intersects $\Ga$ uniformly
transversally}.
\eal
\end{equation}
By Theorem~\ref{thm-H.S} and the considerations following it we can choose
a uniform tubular neighborhood
\begin{equation}\label{S.t}
\chi\sco\Sa(\ve)\ra N(\ve)\times\Sa.
\end{equation}
We write
\hb{D(\ve):=\bigl\{\,(x,y)\in\BR^2\ ;\ x^2+y^2<\ve^2,\ x\geq0\,\bigr\}}.
Then
\begin{equation}\label{S.W}
\wt{U}(\ve):=\chi^{-1}\bigl(D(\ve)\times\Sa\bigr)
\end{equation}
is an open neighborhood of~$\Sa$ in~$M$ contained in~$\Sa(\ve)$. We put
$$
\wh{M}:=M\ssm\Sa
\qb U(\ve):=\wt{U}(\ve)\cap\wh{M}=\wt{U}(\ve)\ssm\Sa.
$$
Furthermore, $r$ and~$\rho$ are given by \eqref{R.r} and \eqref{R.rh},
respectively. Then we define a Riemannian metric~$\wh{g}$ on~$\wh{M}$ by
\begin{equation}\label{S.g}
\wh{g}:=
\left\{
\bal
{}
&g
 &&\ \text{on }M\big\backslash\wt{U}(\ve),\cr
&\chi^*\Bigl(\frac{dx^2+dy^2}{\rho^2(x,y)}+h\Bigr)
 &&\ \text{on }U(\ve).
\eal
\right.
\end{equation}
Note that, see Theorem~\ref{thm-H.Si},
\begin{equation}\label{S.gg}
\wh{g}\sim g\text{ on }M\big\backslash\wt{U}(\ve/3)
\end{equation}
and
$$
\wh{g}=\chi^*\Bigl(\frac{dx^2+dy^2}{x^2+y^2}+h\Bigr)
\text{ on }U(\ve/3).
\npbd
$$
Hence $\whMwhg$~is a Riemannian manifold with~a \emph{wedge singularity
near}~$\Sa$.\po

\par
The following theorem is the basis for our approach. It implies that it
suffices to study transmission problems for membranes without boundary on
urR manifolds.

\begin{theorem}\label{thm-S.M}
$\whMwhg$ is an oriented urR manifold and
\hb{\wh{S}:=S\ssm\Sa} is a membrane in~$\wh{M}$ without boundary.
\end{theorem}

\begin{proof}
(1)
We set
\hb{\dot D(\ve):=D(\ve)\ssm\{0,0\}} and define
\hb{\da\in C^\iy[0,\ve)} by
$$
\rho(x,y)=\da\bigl(r(x,y)\bigr)
\qa (x,y)\in D(\ve).
$$
Then we fix
\hb{\wh{\ve}\in(2\ve/3,\,\ve)}, define a diffeomorphism
$$
s\sco(0,\wh{\ve}\,]\ra\BR_+
\qb r\mt\int_r^{\wh{\ve}}\frac{dt}{\da(t)},
$$
and set
\hb{t:=s^{-1}}. It follows, see \cite[Lemma~5.1]{Ama20a}, that
$$
t^*\Bigl(\frac{dr^2}{\da^2(r)}\Bigr)=ds^2.
$$
We also consider the polar coordinate diffeomorphism
$$
R\sco(0,\wh{\ve}\,)\times[-\pi/2,\,\pi/2]\ra\dot D(\wh{\ve})
\qb (r,\al)\mt r(\cos\al,\sin\al)\ .
$$
Then
$$
R^*(dx^2+dy^2)=dr^2+r^2d\al^2
=\da^2\Bigl(\frac{dr^2}{\da^2}+\frac{r^2}{\da^2}\,d\al^2\Bigr),
$$
that is,
\begin{equation}\label{S.Rg}
R^*\Bigl(\frac{dx^2+dy^2}{\rho^2}\Bigr)
=\frac{dr^2}{\da^2}+\frac{r^2}{\da^2}\,d\al^2.
\end{equation}
Hence
$$
\lda:=R\circ(t\times\id)
\sco(0,\iy)\times[-\pi/2,\,\pi/2]\ra\dot D(\wh{\ve})
$$
is a diffeomorphism satisfying
$$
\lda^*\Bigl(\frac{dx^2+dy^2}{\rho^2}\Bigr)
=(t\times\id)^*R^*\Bigl(\frac{dx^2+dy^2}{\rho^2}\Bigr)
=ds^2+\ba^2(s)d\al^2=:\ga^2,
$$
where
\hb{\ba:=t^*(r/\da)}. By~\eqref{R.rh},
\hb{r/\da=r/(1-\om+r\om)} for
\hb{0<r\leq\wh{\ve}}.
Hence $\ba$~is smooth and
\hb{\ba\sim1}. Thus $\ga$~is a metric on
\hb{N:=(0,\iy)\times[-\pi/2,\,\pi/2]} which is uniformly equivalent to
\hb{g_2=ds^2+d\al^2}. By Examples~\ref{exa-U.ex} \hbox{(a)--(c)},
$$
\bigl(\BR\times[-\pi/2,\,\pi/2],\,ds^2+d\al^2\bigr)
$$
is a urR manifold. From this we deduce, see Remark~\ref{rem-U.R}(a), that
$(N,\ga)$~is a urR manifold on~$(\ol{s},\iy)$ for each
\hb{\ol{s}>0}.

\par
It follows that
$$
w:=(\lda^{-1}\times\id)\circ\chi\sco\bigl(U(\wh{\ve}),\wh{g}\bigr)
\ra(N\times\Sa,\,\ga+h)
$$
is an isometric isomorphism. Hence we derive from Example~\ref{exa-U.ex}(d)
and Remark~\ref{rem-U.R}(a) that $U(\wh{\ve})$~is a urR manifold
on~$U(\ol{\ve})$, where
\hb{\ol{\ve}:=5\ve/6}. Since $\Mg$~is a urR manifold, it is a urR manifold
on
\hb{M\big\backslash\wt{U}(\ve/3)}. Thus it is a consequence of \eqref{S.gg}
that $\whMwhg$~is a urR manifold. The first assertion is now clear.

\par
(2)
Fix
\hb{\ol{\ve}\in(\ve/3,\,\wh{\ve})}. It can be
assumed that \eqref{H.f}~applies with this choice of~$\ol{\ve}$. Set
\hb{\wt{f}_\sa:=t^*f_\sa}. Note that \eqref{H.f}(ii) implies
\begin{equation}\label{S.fpb}
\wt{f}_\sa\sco\bigl[s(\ol{\ve}),\iy\bigr)\ra[-\ol{\rho},\ol{\rho}]
\qa \sa\in\Sa.
\end{equation}
Also note that
\hb{t(s)=c\coc e^{-s}} for
\hb{s\geq s(\ve/3)} and some
\hb{c>0}. Hence
$$
\pl\wt{f}_\sa(s)=-c\coc e^{-s}\pl f_\sa(c\coc e^{-s})
\qa s\geq s(\ve/3).
$$
Thus it follows from \eqref{H.f}(iii) that
\begin{equation}\label{S.df}
\lim_{s\ra\iy}\pl\wt{f}_\sa(s)=0
\qquad\text{$\sa$-unif.},
\npbd
\end{equation}
that is, uniformly with respect to
\hb{\sa\in\Sa}.\po

\par
We write~$\wt{G}_\sa$ for the graph of~$\wt{f}_\sa$ in
\hb{\bigl[s(\ol{\ve}),\iy\bigr)\times[-\pi/2,\,\pi/2]}. We can assume that
\begin{equation}\label{S.nu}
\wt{\nu}_\sa(s)
:=\frac{(\pl\wt{f}_\sa(s),-1)}{(1+|\pl\wt{f}_\sa(s)|^2)^{1/2}}
\qa s\geq s(\ol{\ve}),
\end{equation}
is the positive normal for~$\wt{G}_\sa$ at~$\bigl(s,\wt{f}_\sa(s)\bigr)$
(otherwise replace~$\wt{\nu}_\sa(s)$ by~$-\wt{\nu}_\sa(s)$).
It follows from~\eqref{S.df} that
\begin{equation}\label{S.n}
\lim_{s\ra\iy}\wt{\nu}_\sa(s)=(0,-1)
\qquad\text{$\sa$-unif.}
\end{equation}
From this and \eqref{S.fpb} we deduce that $\wt{G}_\sa$~has a uniform tubular
neighborhood in
\hb{\bigl([s(\ol{\ve}),\iy)\times[-\pi/2,\,\pi/2],\,ds^2+d\al^2\bigr)}
whose width is independent of
\hb{\sa\in\Sa}. It follows from step~(1) that its pull-back by~$w$ is a
uniform tubular neighborhood of~$\wh{S}$
in~$U(\ol{\ve})$. Now the second part of the assertion
is a consequence of Theorem~\ref{thm-H.S}(iv), since, given any
\hb{\da>0},
\,$\wh{g}$~and~$g$ are equivalent on
\hb{M\big\backslash\wt{U}(\da)}.
\end{proof}

\section{Function Spaces}\label{sec-F}
Let $\Mg$ be a Riemannian manifold. We consider the tensor bundles
$$
T_0^1M:=TM
\qb T_1^0M:=T^*M
\qb T_0^0=M\times\BR,
$$
the tangent, cotangent, and trivial bundle, respectively, and
$$
T_\tau^\sa M:=(TM)^{\otimes\sa}\otimes(T^*M)^{\otimes\tau}
\qa \sa,\tau\geq1,
$$
endow~$T_\tau^\sa M$  with the tensor bundle metric
\hb{g_\sa^\tau:=g^{\otimes\sa}\otimes g^{*\,\otimes\tau}},
\,\hb{\sa,\tau\in\BN}, and set\footnote{If $V$~is a vector bundle over~$M$,
then $C^k(V)$~denotes the vector space of $C^k$~sections of~$V$.}
\begin{equation}\label{F.gst}
|a|_{g_\sa^\tau}=\sqrt{(a\sn a)_{g_\sa^\tau}}
:=\sqrt{g_\sa^\tau(a,a)}
\qa a\in C(T_\tau^\sa M).
\end{equation}
By
\hb{\na=\nag} we denote the Levi--Civita connection and interpret it as
covariant derivative. Then, given a smooth function~$u$ on~$M$,
\ \hb{\na^ku\in C^\iy(T_k^0M)} is defined by
\hb{\na^0u:=u},
\ \hb{\na^1u=\na u:=du}, and
\hb{\na^{k+1}u:=\na(\na^ku)} for
\hb{k\in\BN}.

\par
Let
\hb{\ka=(x^1,\ldots,x^m)} be a local coordinate system and set
\hb{\pl_i:=\pl/\pl x^i}. Then
$$
\na^1u=\pl_iu\,dx^i
\qb \na^2u=\na_{\cona ij}u\,dx^i\otimes dx^j
 =(\pl_i\pl_ju-\Ga_{ij}^k\pl_ku)dx^i\otimes dx^j,
$$
where
$$
\Ga_{ij}^k=\frac12\,g^{k\ell}(\pl_ig_{j\ell}+\pl_jg_{i\ell}-\pl_\ell g_{ij})
\qa 1\leq i,j,k\leq m,
$$
are the Christoffel symbols. It follows that
\begin{equation}\label{F.1}
|\na u|_{g_0^1}^2=|\na u|_{g^*}^2=g^{ij}\pl_iu\pl_ju
\end{equation}
and
\begin{equation}\label{F.2}
|\na^2u|_{g_0^2}^2
=g^{i_1j_1}g^{i_2j_2}\na_{\cona i_1i_2}u\na_{\cona j_1j_2}u.
\npbd
\end{equation}
As usual,
\hb{d\vol_g=\sqrt{g}\,dx} is the Riemann--Lebesgue volume element
on~$U_{\coU\ka}$.\po

\bigskip
Let
\hb{\sa,\tau\in\BN}, put
\hb{V:=T_\tau^\sa M}, and write
\hb{\vsdot_V:=\vsdot_{g_\sa^\tau}}. Then $\cD(V)$~is the linear subspace
of~$C^\iy(V)$ of compactly supported sections.

\par
For
\hb{1\leq q<\iy} we set
$$
\|u\|_{L_q(V)}=\|u\|_{L_q\Vg}
:=\Bigl(\int_M|u|_V^q\,d\vol_g\Bigr)^{1/q}.
$$
Then
$$
L_q(V)=L_q\Vg
:=
\bigl(\bigl\{\,u\in L_{1,\loc}(M)\ ;\ \Vsdot_{L_q\Mg}<\iy
\,\bigr\},\ \Vsdot_{L_q\Mg}\bigr)
$$
is the usual Lebesgue space of \hbox{$L_q$~sections} of~$V$. Hence
\hb{L_q\Mg=L_q\Vg} for
\hb{V=T_0^0M=M\times\BR}. If
\hb{k\in\BN}, then
$$
\|u\|_{W_{\coW q}^k(V)}=\|u\|_{W_{\coW q}^k\Vg}
:=\sum_{j=0}^k\big\|\,|\na^jv|_{g_\sa^{\tau+j}}\big\|_{L_q\Mg}
$$
and
$$
\|u\|_{BC^k(V)}=\|u\|_{BC^k\Vg}
:=\sum_{j=0}^k\big\|\,|\na^jv|_{g_\sa^{\tau+j}}\big\|_\iy.
$$
The Sobolev space
\hb{W_{\coW q}^k(V)=W_{\coW q}^k\Vg} is the
completion of $\cD(V)$ in~$L_q(V)$ with respect to the norm~
\hb{\Vsdot_{W_{\coW q}^k(V)}}. If
\hb{k<s<k+1}, the Slobodeckii space $W_{\coW q}^s(V)$ is obtained by real
interpolation:
\begin{equation}\label{F.Ws}
W_{\coW q}^s(V)=W_{\coW q}^s\Vg
:=\bigl(W_{\coW q}^k(V),W_{\coW q}^{k+1}(V)\bigr)_{s-k,q}.
\end{equation}

\par
We also need the time-dependent function spaces
\begin{equation}\label{F.WJ}
W_{\coW q}^{s/\mf2}(M\times J)
:=L_q\bigl(J,W_{\coW q}^s(M)\bigr)\cap W_{\coW q}^{s/2}\bigl(J,L_q(M)\bigr)
\qa 0\leq s\leq2,
\npbd
\end{equation}
Thus
\hb{W_{\coW q}^{0/\mf2}(M\times J)\doteq L_q\bigl(J,L_q(M)\bigr)},
where
\hb{{}\doteq{}}~means `equivalent norms'.\po

\par
By
\hb{BC^k(V)=BC^k\Vg} we denote the Banach space of all
\hb{u\in C^k(V)} for which
\hb{\|u\|_{BC^k(V)}}~is finite, and
\hb{BC:=BC^0}. Then
\begin{equation}\label{F.BJ}
BC^{1/\mf2}(M\times J)
:=C\bigl(J,BC^1(M)\bigr)\cap C^{1/2}\bigl(J,BC(M)\bigr)
\npbd
\end{equation}
with the usual H\"older space~$C^{1/2}$.\po

\par
The following lemma shows that in the Euclidean setting these definitions
return the classical spaces.

\begin{lemma}\label{lem-F.X}
Suppose that
\hb{\BX\in\{\BR^m,\BH^m\}},
\hb{\Mg:=(\BX,g_m)}, and
\hb{V:=\BX\times F} with
\hb{F=\BR^{m^\sa\times m^\tau}\simeq T_\tau^\sa\BX}. Then
$$
W_{\coW q}^s(V)=W_{\coW q}^s\BXF
\qa s\in\BR_+
\qb 1\leq q<\iy,
$$
the standard Sobolev--Slobodeckii spaces, and
$$
BC^k(V)=BC^k\BXF
\qa k\in\BN.
$$
\end{lemma}\po

\begin{proof}
The second assertion is obvious.

\par
If
\hb{k\in\BN}, then the above definition of~$W_{\coW q}^k(V)$ coincides with
the one in~\cite[(VII.1.2.2)]{Ama19a}.
Now the first assertion follows
from~\eqref{F.Ws}, Theorems VII.2.7.4 and VII.2.8.3
in~\cite{Ama19a},
and the fact that the Besov space
\hb{B_q^s=B_{qq}^s} coincides with~$W_{\coW q}^s$ for
\hb{s\notin\BN}.
\end{proof}
Now we suppose that
\begin{equation}\label{F.S}
\bal
{}
&\Mg\text{ is an oriented urR manifold and}\cr
\noalign{\vskip-1\jot}
&S\text{ is a membrane without boundary}.
\eal
\end{equation}
By Theorem~\ref{thm-H.S}(iii) we can chose a uniform tubular neighborhood
\begin{equation}\label{F.pS}
\vp\sco S(\ve)\ra(-\ve,\ve)\times S
\end{equation}
in~$\ci M$. We set
$$
M_+:=\vp^{-1}\bigl([0,\ve)\times S\bigr)
\qb M_-:=\vp^{-1}\bigl((-\ve,0]\times S\bigr)
\qb M_0:=M\big\backslash\ol{S(\ve/2)}.
$$
By
\hb{V_\pm:=V_{M_\pm}} and
\hb{V_0:=V_{M_0}} we denote the restrictions of~$V$ to $M_\pm$ and~$M_0$,
respectively. Then
\hb{\bar W_{\coW q}^s(M\ssm S,\,V)},
\,\hb{s\in\BR_+}, resp.\
\hb{\bar{BC}^k(M\ssm S,\,V)},
\,\hb{k\in\BN}, is the closed linear subspace of
$$
W_{\coW q}^s(V_0)\oplus W_{\coW q}^s(V_+)\oplus W_{\coW q}^s(V_-),
\quad\text{resp.}\quad
BC^k(V_0)\oplus BC^k(V_+)\oplus BC^k(V_-),
$$
consisting of all
\hb{u=u_0\oplus u_+\oplus u_-} satisfying
\hb{(u_0-u_\pm)\sn M_0\cap M_\pm=0}.
Definitions analogous to \eqref{F.WJ} and \eqref{F.BJ} give
the Banach spaces
\hb{\bar W_{\coW q}^{s/\mf2}\bigl((M\ssm S)\times J\bigr)} and
\hb{\bar{BC}^{1/\mf2}\bigl((M\ssm S)\times J\bigr)}, respectively.
Note that
\hb{\bar W_{\coW p}^0(M\ssm S,\,V)=L_p\MV}, since
\hb{\vol_g(S)=0}.

\setcounter{remark}{1}
\begin{remark}\label{rem-F.H}
Let
\hb{\Mg:=(\BR^m,g_m)} and
\hb{S:=\pl\BH^m}. We can set
\hb{\ve=\iy} in \eqref{F.pS} to get
\hb{M_+=\BH^m},
\ \hb{M_-=-\BH^m}, and
\hb{M_0=\es}. Then
$$
\bar W_{\coW q}^s(M\ssm S,\,V)
=W_{\coW q}^s(\BH^m,F)\oplus W_{\coW q}^s(-\BH^m,F)
$$
and
$$
\bar{BC}^k(M\ssm S,\,V)
=BC^k(\BH^m,F)\oplus BC^k(-\BH^m,F),
\npbd
$$
as follows from Lemma~\ref{lem-F.X}.\qed
\end{remark}
Assume
\hb{a\in\bar{BC}(M\ssm S,\,V)}. Then the one-sided limits
$$
\lim_{t\ra0\pm}a\bigl(\ga_p^S(t)\bigr)=:a_\pm(p)
\qa p\in S,
$$
exist and
\hb{a_\pm\in BC(S)}. Hence the \emph{jump across}~$S$,
$$
\eea{a}:=\bigl(p\mt\eea{a}(p)
:=a_+(p)-a_-(p)\bigr)\in BC(S),
$$
is well-defined. Note that $a_\pm$~is the trace of~$a$ on~$S$ `from the
positive/negative side of~$S$'.

\par
Let
\hb{u\in\bar{BC}^1(M\ssm S)}. Then
\hb{u\circ\ga_p^S\in\bar{BC}^1\bigl((-\ve,\ve)\ssm\{0\}\bigr)}. We set
$$
\frac{\pl u}{\pl\nu_S}(q)
:=\pl_1(u\circ\vp^{-1})(\tau,p)
\qa q\in (M_+\cup M_-)\ssm S,
$$
for
\hb{\vp(q)=(\tau,p)\in(-\ve,\ve)\times S} with
\hb{\tau\neq0}. Thus
\hb{\pl u/\pl\nu_S} is the \emph{normal derivative} of~$u$ in
\hb{(M_+\cup M_-)\ssm S}, that is, the derivative along the normal
geodesic~$\ga_p^S$. Hence
$$
\frac{\pl u}{\pl\nu_S}(q)
=\bigl\dl du(q),\dot\ga_p^S(\tau)\bigr\dr
=\bigl(\dot\ga_p^S(\tau)\bsn\grad u(q)\bigr)_{g(q)}.
$$
Consequently, the \emph{jump of the normal derivative},
$$
\Beea{\displaystyle\frac{\pl u}{\pl\nu_S}}
=\beea{(\nu_S\sn\grad u)}
\in BC(S)
\npbd
$$
is also well-defined.\po

\section{The Parabolic Problem on Manifolds}\label{sec-L}
We presuppose \eqref{F.S} and assume
\begin{equation}\label{L.a}
\bal
{\rm(i)}\quad
&a\in\bar{BC}^{1/\mf2}\bigl((M\ssm S)\times J\bigr).\cr
{\rm(ii)}\quad
&a\geq\ul{\al}>0,
\eal
\end{equation}
where
\hb{\ul{\al}\leq1}. Then
$$
\cA u:=-\tdiv(a\grad u).
$$

\par
Fix
\hb{\da\in C\bigl(\Ga,\{0,1\}\bigr)}. Then
\hb{\Ga_j:=\da^{-1}(j)},
\,\hb{j=0,1}, is open and closed in~$\Ga$ and
\hb{\Ga_0\cup\Ga_1=\Ga}. Either $\Ga$, $\Ga_0$, or~$\Ga_1$ may be empty.
In such a case all references to these empty sets have to be disregarded.
Recall that $\ga$~denotes the trace operator on~$\Ga$ (for any manifold).

\par
We introduce an operator
\hb{\cB=(\cB^0,\cB^1)} on~$\Ga$ by
\hb{\cB^0u=\ga_{\Ga_0}u}, the Dirichlet boundary operator, on~$\Ga_0$, and
a~Neumann boundary operator
$$
\cB^1u:=\bigl(\nu\bsn\ga_{\Ga_1}(a\grad u)\bigr)\text{ on }\Ga_1.
$$

\par
On~$S$ we consider the transmission operator
\hb{\cC=(\cC^0,\cC^1)}, where
$$
\cC^0u:=\eea{u}
\qb \cC^1u:=\beea{(\nu_S|a\grad u)}.
\npbd
$$
Note that
\hb{\beea{(\nu_S\sn a\grad u)}} equals
\hb{\eea{a\pl_{\nu_S}u}} and not
\hb{\beea{\pl_{\nu_S}(au)}}.\po

\par
Of concern in this paper is the inhomogeneous linear transmission
problem
\begin{equation}\label{L.T}
\bal
\pl_tu+\cA u    &=f     &&\text{ on }(M\ssm S)\times J,\cr
\cB u           &=\vp   &&\text{ on }\Ga\times J,\cr
\cC u           &=\psi  &&\text{ on }S\times J,\cr
\ga_0u          &=u_0   &&\text{ on }(M\ssm S)\times\{0\}.
\eal
\end{equation}\po

\par
We assume that
\begin{equation}\label{L.ass}
1<p<\iy,\ p\notin\{3/2,\,3\},
\ \text{\eqref{F.S} and \eqref{L.a} are satisfied}.
\end{equation}

\par
For abbreviation we set, for
\hb{1<q<\iy},
$$
\bar W_{\coW q}^{k/\mf2}
:=\bar W_{\coW q}^{k/\mf2}\bigl((M\ssm S)\times J\bigr)
\qa k=0,2,
$$
and introduce the trace spaces
$$
\bal
\pl W_{\coW q}
&:=W_{\coW q}^{(2-1/q)/\mf2}(\Ga_0\times J)
 \oplus W_{\coW q}^{(1-1/q)/\mf2}(\Ga_1\times J),\cr
\pl_SW_{\coW q}
&:=W_{\coW q}^{(2-1/q)/\mf2}
 (S\times J)\oplus W_{\coW q}^{(1-1/q)/\mf2}(S\times J),
\eal
$$
and
$$
\ga_0\bar W_{\coW q}
:=\bar W_{\coW q}^{2-2/q}(M\ssm S).
$$
As a rule, we often drop the index~$q$ if
\hb{q=p}. Thus
\hb{\bar W^{2/\mf2}=\bar W_{\coW p}^{2/\mf2}},
\ \hb{\pl W=\pl W_{\coW p}}, etc. Finally,
$$
\pl_{\cB,\cC}W
=[\pl W\oplus\pl_SW\oplus\ga_0\bar W]_{\cB,\cC}
$$
is the closed linear subspace of
\hb{\pl W\oplus\pl_SW\oplus\ga_0\bar W} consisting of all
$(\vp,\psi,u_0)$ satisfying the compatibility conditions
$$
\bal
\cB^0u_0
&=\vp^0(0),     &\quad \cC^0u_0     &=\psi^0(0),
&&\quad\text{if}\quad
&3/2   &<p<3,\cr
\cB(0)u_0
&=\vp(0),       &\quad \cC(0)u_0    &=\psi(0),
&&\quad\text{if}\quad
&3     &<p<\iy,
\eal
\npbd
$$
where
\hb{\vp=(\vp^0,\vp^1)} and
\hb{\psi=(\psi^0,\psi^1)}. It follows from the anisotropic trace theorem
(\cite[Example VIII.1.8.6]{Ama19a}) that $\pl_{\cB,\cC}W$~is well-defined.

\par
Given Banach spaces $E$ and~$F$, we  write~$\Lis\EF$ for the set
of all isomorphisms in~$\cL\EF$, the Banach space of
continuous linear maps from~$E$ into~$F$.

\par
Now we can formulate the following maximal regularity theorem for
problem~\eqref{L.T}. Its proof, which needs considerable preparation,
is found in Section~\ref{sec-P}.

\begin{theorem}\label{thm-L.MR}
Let \eqref{L.ass} be satisfied. Then
$$
\bigl(\pl_t+\cA,\,(\cB,\cC,\ga_0)\bigr)
\in\Lis\bigl(\bar W_{\coW p}^{(2,1)},
L_p(J,L_p(M))\times\pl_{\cB,\cC}W_{\coW p}\bigr).
$$
\end{theorem}

\section{The Uniform Lopatinskii--Shapiro Condition}\label{sec-E}
In the proof of Theorem~\ref{thm-L.MR} we need to consider systems of
elliptic boundary value problems. For this we have to be precise on the
concept of uniform ellipticity.

\par
Let $\Mg$ be any Riemannian manifold. We consider a general second order
linear differential operator on~$M$,
\begin{equation}\label{E.A}
\cA u:=-a\btdot\na^2u+a_1\btdot\na u+a_0u
\end{equation}
with
\hb{u=(u^1,\ldots,u^n)} and
$$
a_i\in C(T_0^iM)^{n\times n}
\qa i=0,1,2
\qb a_2:=a.
$$
Here
\hb{\na^iu=(\na^iu^1,\ldots,\na^iu^n)} so that, for example,
$$
a\btdot\na^2u=(a_s^1\btdot\na^2u^s,\ldots,a_s^n\btdot\na^2u^s),
$$
where $s$~is summed from~$1$ to~$n$ and
\hb{{}\btdot{}}~denotes complete contraction, that is, summation over all
twice occurring indices in any local coordinate representation.

\par
The (principal) symbol~$\gss\cA$ of~$\cA$ is the
\hb{(n\times n)}-matrix-valued function defined by
$$
\gss\cA(p,\xi):=a(p)\btdot(\xi\otimes\xi)
\qa p\in M
\qb \xi\in T_p^*M.
$$
Then $\cA$~is \emph{uniformly normally elliptic} if there exists an
`ellipticity constant'
\hb{\ul{\al}\in(0,1)} such that
$$
\sa\bigl(\gss\cA(p,\xi)\bigr)\is[\Re z\geq\ul{\al}]
=\{\,z\in\BC\ ;\ \Re z\geq\ul{\al}\,\}
\npbd
$$
for all
\hb{p\in M} and
\hb{\xi\in T_p^*M} with
\hb{|\xi|_{g^*(p)}^2=1}, where $\sa(\cdot)$~denotes the spectrum.\po

\par
Suppose
\hb{\Ga\neq\es} and
\hb{\cB=(\cB^1,\ldots,\cB^n)} is a linear boundary operator of order
at most~$1$. More precisely, we assume that there is
\hb{k\in\{0,\ldots,n\}} such that
$$
\cB^ru=
\left\{
\bal
{}
&b_0^r\btdot\ga u,
&\quad 1    &\leq r\leq k,\cr
&b_1^r\btdot\ga\na u+b_0^r\btdot\ga u,
&\quad k+1  &\leq r\leq n,\cr
\eal
\right.
$$
with
$$
b_i^r\in C(T_0^i\Ga)^n
\qa 1\leq r\leq n
\qb i=0,1.
$$
Then the (principal) symbol of~$\cB$ is the
\hb{(n\times n)}-matrix-valued function~$\gss\cB$ given by
$$
\gss\cB^r(q,\xi):=
\left\{
\bal
{}
&b_0^r(q),\cr
&b_1^r(q)\btdot\xi,
\eal
\right.
\qquad q\in\Ga
\qb \xi\in T_q^*M.
$$
Observe that
\hb{X\btdot\om=\dl\om,X\dr} if $X$~is a vector, $\om$~a covector field, and
\hb{\pw} the duality pairing.

\par
We denote by
\hb{\nu^\flat\in T_\Ga^*M} the inner conormal on~$\Ga$ defined in local
coordinates by
\hb{\nu^\flat=g_{ij}\nu^jdx^i}. Given
\hb{q\in\Ga}, we write~$\BB(q)$ for the set of all
\begin{equation}\label{E.Bq}
\bal
{}
&(\xi,\lda)\in T_q^*M\times[\Re z\geq0]\text{ satisfying}\cr
&\xi\perp\nu^\flat(q)\text{ and }|\xi|^2+|\lda|=1.
\eal
\end{equation}
Then, if
\hb{(\xi,\lda)\in\BB(q)}, we introduce linear differential  operators
on~$\BR$ by
\begin{equation}\label{E.AB}
\bal
A(\pl;q,\xi,\lda)
&:=\lda+\gss\cA\bigl(q,\,\xi+\imi\nu^\flat(q)\pl\bigr),\cr
B(\pl;q,\xi,\lda)
&:=\gss\cB\bigl(q,\,\xi+\imi\nu^\flat(q)\pl\bigr),
\eal
\npbd
\end{equation}
where
\hb{\imi=\sqrt{-1}}.\po

\par
As usual, $C_0(\BR_+,\BC^n)$ is the closed linear subspace
of $BC(\BR_+,\BC^n)$ consisting of the functions that vanish at infinity.

\par
Suppose $\cA$~is uniformly normally elliptic. Then it follows that the
homogeneous problem
\begin{equation}\label{E.Aa}
A(\pl;q,\xi,\lda)v=0\text{ on }\BR
\end{equation}
has for each
\hb{q\in\Ga} and
\hb{(\xi,\lda)\in\BB(q)} precisely~$n$ linearly independent solutions
whose restrictions to~$\BR_+$ belong to~$C_0(\BR_+,\BC^n)$.
We denote their span by $C_0(q,\xi,\lda)$. It is an \hbox{$n$-dimensional}
linear subspace of $C_0(\BR_+,\BC^n)$.

\par
Now we consider the initial value problem on the half-line:
\begin{equation}\label{E.LS}
\bal
A(\pl;q,\xi,\lda)v      &=0\text{ on }\BR_+,\cr
B(\pl;q,\xi,\lda)v(0)   &=\eta\in\BC^n.
\eal
\end{equation}
Then $\AB$~satisfies the \emph{uniform parameter-dependent
Lopatinskii-Shapiro} (LS) \emph{conditions}
if problem~\eqref{E.LS} has for each
\hb{\eta\in\BC^n} a~unique solution
\begin{equation}\label{E.v}
v=R(q,\xi,\lda)\eta\in C_0(q,\xi,\lda)
\npbd
\end{equation}
and
\begin{equation}\label{E.R}
\|R(q,\xi,\lda)\|_{\cL(\BC^n,C_0(\BR_+,\BC^n))}\leq c,
\npbd
\end{equation}
unif.\ w.r.t.\
\hb{q\in\Ga} and
\hb{(\xi,\lda)\in\BB(q)}.\po

\par
The basic feature, which distinguishes the above definition from the usual
form of the LS~condition, is the requirement of the uniform
bound~\eqref{E.R}. Without this requirement the LS~condition is much
simpler to
formulate (e.g.,~\cite{Ama90a}, \cite{Ama93a}, \cite{DHP03a},
\cite{DHP07a}, \cite{PrS16a}, for example) and to verify.

\par
It is known that the LS~condition is equivalent to the parameter-dependent
version of the complementing condition of
S.~Agmon, A.~Douglis, and L.~Nirenberg~\cite{ADN64a} (see, for example,
\cite[VII\S9]{LSU68a} or \cite[Section~10.1]{WRL95a}). Using this version,
it is possible to define a uniform complementing condition which is
equivalent to~\eqref{E.R} (see \cite{Ama84b} and~\cite{Ama85a}).
However, that condition is even more difficult to verify in concrete
situations. We refer to~\cite{AmaVolIII21a} for a detailed exposition of
all these facts. It should be noted that the uniformity
condition~\eqref{E.R} is fundamental for the following, since we will have
to work with infinitely many linear model problems.

\section{Model Cases}\label{sec-M}
For the proof of Theorem~\ref{thm-L.MR} we have to understand the model
cases to which problem~\eqref{L.T} reduces in local coordinates.

\par
Until further notice, it is assumed that
$$
\bal
\bt\quad
&\text{assumption \eqref{L.ass} applies}.\cr
\bt\quad
&\gK\text{ is an $S$-adapted ur atlas for }M.
\eal
$$
By Remark~\ref{rem-U.R}(b) we can assume that
\hb{\diam(U_{\coU\ka})<\ve/2} for
\hb{\ka\in\gK_S} where $\ve$~is the width of the tubular neighborhood
of~$S$.

\par
We can choose a family
\hb{\{\,\pi_\ka,\chi\ ;\ \ka\in\gK\,\}} with the following properties:
\begin{equation}\label{M.LS}
\bal
{\rm(i)}\quad
&\pi_\ka\in\cD\bigl(U_{\coU\ka},[0,1]\bigr)\text{ for $\ka\in\gK$ and }
 {\textstyle\sum_\ka}\pi_\ka^2(p)=1\text{ for }p\in M.\cr
{\rm(ii)}\quad
&\|\ka_*\pi_\ka\|_{k,\iy}\leq c(k),\ \ka\in\gK,\ k\in\BN.\cr
{\rm(iii)}\quad
&\chi\in\cD\bigl((-1,1)^m,[0,1]\bigr)
 \text{ with }\chi\sn\supp(\ka_*\pi_\ka)=1\text{ for }\ka\in\gK.
\eal
\end{equation}
(See Lemma~3.2 in~\cite{Ama12b} or~\cite{AmaVolIII21a}). We fix an
\hb{\wt{\om}\in\cD\bigl((-1,1)^m,[0,1]\bigr)} which is equal to~$1$ on
$\supp(\chi)$. Then
$$
g_\ka:=\wt{\om}\ka_*g+(1-\wt{\om})g_m
$$
is a Riemannian metric on~$\BR^m$ such that
\begin{equation}\label{M.ka}
g_\ka\sim g_m
\qa \ka\in\gK,
\end{equation}
and
\begin{equation}\label{M.g}
\|g_\ka\|_{k,\iy}\leq c(k)
\qa \ka\in\gK
\qb k\in\BN.
\end{equation}
This follows from \eqref{U.g}. Furthermore,
\begin{equation}\label{M.a}
a_\ka:=\wt{\om}\ka_*a+1-\wt{\om}.
\end{equation}
Note that
\begin{equation}\label{M.aa}
a_\ka\geq\ul{\al}
\qa \ka\in\gK.
\end{equation}

\par
We write
\hb{\grad_\ka:=\grad_{g_\ka}} and
\hb{\tdiv_\ka:=\tdiv_{g_\ka}} for
\hb{\ka\in\gK}. Then
$$
\cA_\ka u:=-\tdiv_\ka(a_\ka\grad_\ka u)
\qa u\in Q_\ka^m.
$$
Let
\hb{\da_\ka:=\ka_*\da}. Then
$$
\cB_\ka u
:=\da_\ka\bigl(\nu_\ka\bsn\ga(a_\ka\grad_\ka u)\bigr)_\ka+(1-\da_\ka)\ga u
\qa \ka\in\gK_\Ga,
$$
where $\nu_\ka$~is the inner normal on~$\pl\BH^m$ with respect to
$(\BH^m,g_\ka)$, and
\hb{\prsn_\ka=g_\ka}. If
\hb{\ka\in\gK_S}, then
$$
\cC_\ka u:=\Bigl(\eea{u},\beea{(\nu_\ka\sn a_\ka\grad_\ka u)_\ka}\Bigr)
\text{ on }\pl\BH^m.
$$
Using these notations, we consider the three model problems:
\begin{equation}\label{M.P1}
\pl_tu+\cA_\ka u=f_\ka\text{ on }\BR^m\times J,
\end{equation}
and
\begin{equation}\label{M.P2}
\bal
\pl_tu+\cA_\ka u&=f_\ka     &&\text{ on }\BH^m\times J,\cr
\cB_\ka u       &=\vp_\ka   &&\text{ on }\pl\BH^m\times J,
\eal
\end{equation}
and
\begin{equation}\label{M.P3}
\bal
\pl_tu+\cA_\ka u&=f_\ka     &&\text{ on }(\BR^m\ssm\pl\BH^m)\times J,\cr
\cC_\ka u       &=\psi_\ka  &&\text{ on }\pl\BH^m\times J.
\eal
\end{equation}

\par
In the following two sections we prove that each one of them,
complemented by appropriate initial and compatibility conditions, enjoys a
maximal regularity result, unif.\ w.r.t.~$\ka$.

\section{Continuity}\label{sec-C}
First we note that
\begin{equation}\label{C.g}
\sqrt{g_\ka}\sim1
\qa \ka\in\gK,
\end{equation}
and, given
\hb{k\in\BN},
\begin{equation}\label{C.Nd}
\sum_{i=0}^k|\naka^iu|\sim\sum_{|\al|\leq k}|\pa u|
\qa \ka\in\gK
\qb u\in C^k(Q_\ka^m),
\npbd
\end{equation}
with
\hb{\naka u:=\ka_*\na\ka^*u}
(cf.~\cite[Lemma~3.1]{Ama12b} or~\cite{AmaVolIII21a}).

\par
We set
\begin{equation}\label{C.X}
\BX_\ka:=
\left\{
\bal
{}
&\BR^m,
 &&\quad \text{if }\ka\in\gK_0:=\gK\ssm(\gK_\Ga\cup\gK_S),\cr
&\BH^m,
 &&\quad \text{if }\ka\in\gK_\Ga,\cr
&\BR^m\ssm\pl\BH^m,
 &&\quad \text{if }\ka\in\gK_S.
\eal
\right.
\end{equation}
Then
$$
W_{\coW\ka}^s:=W_{\coW p}^s(\BX_\ka,g_\ka)
\qa \ka\in\gK_0\cup\gK_\Ga,
$$
and
$$
\bar W_{\coW\ka}^s:=\bar W_{\coW p}^s(\BR^m\ssm\pl\BH^m,\,g_\ka)
\qa \ka\in\gK_S,
$$
where
\hb{0\leq s\leq2}. For the sake of a unified presentation,
$$
\sW_{\coW\ka}^s:=
\left\{
\bal
{}
&W_\ka^s,
 &&\quad \text{if }\ka\in\gK\ssm\gK_S,\cr
&\bar W_{\coW\ka}^s,
 &&\quad \text{if }\ka\in\gK_S.
\eal
\right.
$$
If
\hb{\BX\in\{\BR^m,\BH^m\}}, then
\hb{W_{\coW p}^s(\BX):=W_{\coW p}^s(\BX,g_m)}. It is a consequence of
\eqref{C.g} and \eqref{C.Nd} that
\begin{equation}\label{C.Wk}
\sW_{\coW\ka}^k\doteq\sW_{\coW p}^k(\BX_\ka)
\qquad\text{$\gK$-unif.},
\npbd
\end{equation}
where
\hb{{}\doteq{}}~stands for `equal except for equivalent norms'.\po

\par
Since
$$
\sW_{\coW p}^s(\BX)
=\bigl(\sW_{\coW p}^k(\BX),\sW_{\coW p}^{k+1}(\BX)\bigr)_{s-k,p}
\qa k<s<k+1,
$$
(cf.~\cite[Theorems VII.2.7.4 and VII.2.8.3, as well as
(VII.3.6.3)]{Ama19a}),
it follows from definition~\eqref{F.Ws} and from \eqref{C.Wk} that
\begin{equation}\label{C.W}
\sW_{\coW\ka}^s\doteq\sW_{\coW p}^s(\BX_\ka)
\qquad\text{$\gK$-unif.}
\end{equation}
Due to \eqref{C.g} and \eqref{C.Nd} we get, with an analogous definition
of~$\sBC$,
\begin{equation}\label{C.B}
\sBC_\ka^k
:=\sBC^k(\BX_\ka,g_\ka)
\doteq\sBC^k(\BX_\ka):=\sBC^k(\BX_\ka,g_m)
\qquad\text{$\gK$-unif.}
\end{equation}
Using this, \eqref{F.WJ}, and \eqref{F.BJ}, we infer that
\begin{equation}\label{C.WW}
\sW_{\coW\ka}^{s/\mf2}\doteq\sW_{\coW p}^{s/\mf2}(\BX_\ka\times J)
\qb \sBC_\ka^{k/\mf2}\doteq\sBC^{k/\mf2}(\BX_\ka\times J)
\qquad \text{$\gK$-unif}.
\end{equation}\po

\par
First we note that \eqref{U.K}, \eqref{L.a}, \eqref{M.a}, and
\eqref{C.Nd} imply
\begin{equation}\label{C.ak}
a_\ka\in\sBC_\ka^{1/\mf2}
\qquad\text{$\gK$-unif}.
\end{equation}
In local coordinates,
\hb{\grad u=g^{ij}\pl_ju\,\pl/\pl x^i}. Using this, \eqref{U.g}, and
\eqref{C.ak} we deduce that
\begin{equation}\label{C.a}
\|\grad_\ka a_\ka\|_{\sBC_\ka(T\BX_\ka\times J)}\leq c
\qquad\text{$\gK$-unif.}
\end{equation}

\par
Given a vector field
\hb{Y=Y^i\pl/\pl x^i}, it holds
\hb{\tdiv Y=\sqrt{g}^{\kern1pt-1}\pl_i\bigl(\sqrt{g}\,Y^i\bigr)}.
By this and the above it is verified that
\begin{equation}\label{C.Aku}
\cA_\ka\in\cL(\sW_{\coW\ka}^{2/\mf2},\sW_{\coW\ka}^{0/\mf2})
\qquad\text{$\gK$-unif.}
\end{equation}

\par
Now we consider Sobolev--Slobodeckii spaces on
\hb{\pl\BH^m\simeq\BR^{m-1}}. We set
\hb{g_\ka^\fdot:=g_{\ka\pl\BH^m}} for
\hb{\ka\in\gK_\Ga}. Then
\begin{equation}\label{C.diW}
\pl_iW_{\coW\ka}
:=W_{\coW p}^{(2-i-1/p)/\mf2}(\pl\BH^m\times J,\,g_\ka^\fdot+dt^2)
\qa i=0,1,
\npbd
\end{equation}
and
$$
\pl W_{\coW\ka}:=(1-\da_\ka)\pl_0W_{\coW\ka}+\da_\ka\pl_1W_{\coW\ka}
\qa \ka\in\gK_\Ga.
$$

\par
Suppose
\hb{0<\sa<s<1}. Then
\begin{equation}\label{C.BB}
\bal
BC^{1/\mf2}(\BR^{m-1}\times J)
&=C\bigl(J,BC^1(\BR^{m-1})\bigr)\cap C^{1/2}\bigl(J,BC(\BR^{m-1})\bigr)\cr
&=C\bigl(J,BC^1(\BR^{m-1})\bigr)\cap C^{1/2}\bigl(J,\BUC(\BR^{m-1})\bigr)\cr
&\hr B\bigl(J,\BUC^s(\BR^{m-1})\bigr)
 \cap C^{s/2}\bigl(J,\BUC(\BR^{m-1})\bigr)\cr
&\doteq\BUC^{s/\mf2}(\BR^{m-1}\times J)
 \hr b_\iy^{\sa/\mf2}(\BR^{m-1}\times J),
\eal
\end{equation}
where the $\BUC^\rho$~are the usual H\"older spaces and $b_\iy^{\sa/\mf2}$
is an anisotropic little Besov space. Indeed, the first
embedding follows from the mean value theorem and by using the localized
H\"older norm (cf.~\cite[(VII.3.7.1)]{Ama19a}).
For the norm equivalence we refer to definition~(VII.3.6.4) and
Remark~VII.3.6.4. The last embedding is implied by Lemma~VII.2.2.3 and
Theorem~VII.7.3.4. By Theorem~VII.2.7.4 in~\cite{Ama19a},
\begin{equation}\label{C.int}
\bal
{}
&b_\iy^{\sa/\mf2}(\BR^{m-1}\times J)\cr
&\qquad{}
 \doteq\bigl(\BUC^{2/\mf2}(\BR^{m-1}\times J),\,
 \BUC^{0/\mf2}(\BR^{m-1}\times J)\bigr)_{\sa/2,\iy}^0.
\eal
\end{equation}
We deduce from \eqref{U.g} and \eqref{C.Nd} that
$$
\BUC_\ka^{s/\mf2}(\pl\BH^m\times J)
:=\BUC^{s/\mf2}(\pl\BH^m\times J,\,g_\ka^\fdot+dt^2)
\doteq\BUC^{s/\mf2}(\BR^{m-1}\times J)
$$
$\gK_\Ga$-unif. Now it follows from \eqref{C.BB} and \eqref{C.int} that
$$
BC_\ka^{1/\mf2}(\pl\BH^m\times J)
\hr b_{\iy,\ka}^{\sa/\mf2}(\pl\BH^m\times J)
\qquad\text{$\gK_\Ga$-unif.}
$$
Since, trivially,
\hb{\ga\in\cL\bigl(BC^{1/\mf2}(\BH^m\times J),
   \,BC^{1/\mf2}(\pl\BH^m\times J)\bigr)}, it is now clear that
\begin{equation}\label{C.ab}
\ga a_\ka\in b_{\iy,\ka}^{\sa/\mf2}(\pl\BH^m\times J)
\qquad\text{$\gK_\Ga$-unif.}
\end{equation}

\par
In local coordinates
\begin{equation}\label{C.nu}
\nu_\ka=\frac{g_\ka^{1i}}{\sqrt{g_\ka^{11}}}\,\frac\pl{\pl x^i}.
\end{equation}
Hence
\hb{\da_\ka\cB_\ka u=\da_\ka b_\ka^i\ga\pl_iu
   =\da_\ka\ga a_\ka\nu_\ka^i\ga\pl_iu_\ka},
where it follows from \eqref{U.g},
\eqref{M.aa}, and
\hb{\|a_\ka\|_\iy\leq c} that
\begin{equation}\label{C.b1}
1/c\leq b_\ka^1=\ga a_\ka\sqrt{g_\ka^{11}}\leq c
\end{equation}
and, from \eqref{C.ab},
$$
\|b_\ka^i\|_{b_{\iy,\ka}^{\sa/\mf2}(\pl\BH^m\times J)}\leq c
\qa 1\leq i\leq m,
$$
for
\hb{\ka\in\gK_\Ga}. Thus it is a consequence of \eqref{C.ab},
\eqref{C.b1}, and the boundary operator retraction theorem
\cite[Theorem~VIII.2.2.1]{Ama19a}
that\addtocounter{footnote}{1}\footnotetext{An operator
\hb{r\in\cL\EF} is a retraction if it has a continuous right inverse,
a~coretraction~$r^c$. Then $(r,r^c)$ is an
\emph{\hbox{r-c} pair for}~$\EF$.}
\begin{equation}\label{C.RB}
\addtocounter{footnote}{-1}
\cB_\ka\text{ is a $\gK_\Ga$-uniform retraction\footnotemark\
from $W_{\coW\ka}^{2/\mf2}$ onto }\pl W_{\coW\ka}.
\end{equation}
Clearly, `\hbox{$\gK_\Ga$-uniform}' means that there exists a
coretraction~$\cB_\ka^c$ such that $\|\cB_\ka\|$ and~$\|\cB_\ka^c\|$ are
\hbox{$\gK_\Ga$-uniformly} bounded.

\par
Obviously,
\begin{equation}\label{C.WSJ}
\pl_SW_{\coW\ka}:=\pl_0W_{\coW\ka}\oplus\pl_1W_{\coW\ka}
\qa \ka\in\gK_S.
\end{equation}
If we replace in the preceding arguments the boundary
operator retraction argument by Theorem~VIII.2.3.3 of~\cite{Ama19a},
we find that
\begin{equation}\label{C.Rc}
\cC_\ka\text{ is a $\gK_S$-uniform retraction
from $\bar W_{\coW\ka}^{2/\mf2}$ onto }\pl_S W_{\coW\ka}.
\end{equation}

\par
It follows from \eqref{C.W} that
\begin{equation}\label{C.dW}
\ga_0\sW_{\coW\ka}\doteq\ga_0\sW_{\coW p}(\BX_\ka)
\qa \ka\in\gK.
\end{equation}
The anisotropic trace theorem (\cite[Corollary~VII.4.6.2,
Theorems VIII.1.2.9 and~VIII.1.3.1]{Ama19a}) implies that
\begin{equation}\label{C.g0}
\ga_0\in\cL\bigl(W_{\coW p}^{2/\mf2}(\BX\times J),B_p^{2-2/p}(\BX)\bigr)
\qa \BX\in\{\BR^m,\BH^m\},
\end{equation}
is a retraction. Using Theorems VII.2.7.4 and~VII.2.8.3,
definition~VII.3.6.3 and Remark~VII.3.6.4 of~\cite{Ama19a}, we get
\begin{equation}\label{C.BW}
B_p^{2-2/p}(\BX)\doteq W_{\coW p}^{2-2/p}(\BX).
\end{equation}
Now we infer from \eqref{C.WW}, \hbox{\eqref{C.dW}--\eqref{C.BW}},

and \eqref{C.W} that
\begin{equation}\label{C.0}
\ga_0\text{ is a $\gK$-uniform retraction
from $\sW_{\coW\ka}^{2/\mf2}$ onto }\ga_0\sW_{\coW\ka}.
\end{equation}

\section{Maximal Regularity}\label{sec-MR}
First we rewrite $\AB$ in terms of covariant derivatives.
For this we define
$$
a^\nat\in\bar{BC}^{1/\mf2}\bigl(T_0^2(M\ssm S)\times J\bigr)
$$
in local coordinates by
$$
a^\nat:=ag^{ij}\frac\pl{\pl x^i}\otimes\frac\pl{\pl x^j}.
$$
Then we get
\begin{equation}\label{MR.A}
\cA u=-a\Da u-(\grad a\sn\grad u)
=-a^\nat\btdot\na^2u-(\grad a)\btdot\na u,
\end{equation}
where $\Da$~is the Laplace--Beltrami operator
(e.g.,~\cite[(2.4.10)]{Tay96a}). Consequently,
\begin{equation}\label{MR.sA}
\gss\cA(q,t,\xi)=a(q,t)\,|\xi|^2_{g^*(q)}
\qa q\in M\ssm S
\qb \xi\in T_q^*(M\ssm S)
\qb t\in J.
\end{equation}

\par
For the boundary operator we find
\begin{equation}\label{MR.B}
\cB^1u=\ga a(\nu^\flat\sn\ga\na u)_{g^*}.
\end{equation}
Hence
\begin{equation}\label{MR.sB}
\gss\cB^1u(q,t,\xi)=a(q,t)\bigl(\nu^\flat(q)\bsn\xi\bigr)_{g^*(q)}
\qa q\in\Ga
\qb \xi\in T_q^*M
\qb t\in J.
\end{equation}
Clearly, these formulas apply to any oriented Riemannian manifold, thus to
$(\pl\BH^m,g_\ka)$,
\ \hb{\ka\in\gK_\Ga}.

\par
It follows from \eqref{M.aa} and \eqref{MR.sA} that
$$
\gss\cA_\ka(x,t,\xi)
=a_\ka(x,t)\,|\xi|_{g_\ka^*(x)}^2\geq\ul{\al}\,|\xi|_{g_\ka^*(x)}^2
$$
for
\,\hb{x\in\BX_\ka},
\,\hb{\xi\in T_x^*\BX_\ka},
\,\hb{t\in J}, and
\hb{\ka\in\gK}. Hence
\begin{equation}\label{MR.ne}
\cA_\ka\text{ is uniformly normally elliptic, unif.\ w.r.t.\ }\ka\in\gK
\text{ and }t\in J.
\end{equation}

\par
We begin with the full-space problem.

\begin{proposition}\label{pro-MR.1}
It holds
$$
(\pl_t+\cA_\ka,\,\ga_0)
\in\Lis(W_{\coW\ka}^{2/\mf2},\,W_{\coW\ka}^{0/\mf2}\times\ga_0W_{\coW\ka})
\qquad\text{$\gK_0$-unif.},
$$
that is,
$$
\|(\pl_t+\cA_\ka,\,\ga_0)\|
+\|(\pl_t+\cA_\ka,\,\ga_0)^{-1}\|\leq c
\qa \ka\in\gK_0.
$$
\end{proposition}

\begin{proof}
It is obvious from \eqref{C.Aku} and \eqref{C.0} that
$$
(\pl_t+\cA_\ka,\,\ga_0)
\in\cL(W_{\coW\ka}^{2/\mf2},\,W_{\coW\ka}^{0/\mf2}\times\ga_0 W_{\coW\ka})
\qquad\text{$\gK_0$-unif.}
$$
Due to \eqref{MR.ne}, the assertion now follows from Corollary~9.7
in~\cite{AHS94a} and Theorem~III.4.10.8 in~\cite{Ama95a} and (the proof of)
Theorem~7.1 in~\cite{Ama04a}. (See \cite{AmaVolIII21a} for a different
demonstration.)
\end{proof}
Next we study the case where
\hb{\ka\in\gK_\Ga}. For this we first establish the validity of the
uniform LS condition. Henceforth, it is always assumed that
\begin{equation}\label{MR.Z}
\za=(x,\xi,\lda)\text{ with }x\in\pl\BH^m\text{ and }(\xi,\lda)\in\BB(x).
\end{equation}
We fix any
\hb{t\in J} and omit it from the notation. The reader will easily check
that all estimates are uniform with respect to
\hb{t\in J}. From \eqref{MR.sA} we see that the first equation in
\eqref{E.Aa} has the form
\begin{equation}\label{MR.vR}
\ddot v=\rho_\ka^2(\za)v\text{ on }\BR,
\end{equation}
where
\begin{equation}\label{MR.rh}
\rho_\ka(\za)
:=\sqrt{\frac\lda{a_\ka(x)}+|\xi|_{g_\ka^*(x)}^2}\in\BC
\npbd
\end{equation}
with the principal value of the square root.\po

\par
Suppose
\hb{|\xi|_{g_\ka^*(x)}^2\leq1/2}. Then
\hb{\za\in\BB(x)} implies
\hb{\rho_\ka^2(\za)\geq1/2a_\ka(x)}. Otherwise,
\hb{\rho_\ka^2(\za)\geq1/2}. Thus, since
\hb{\|a_\ka\|_\iy\leq c}, we find a
\hb{\ba>0} such that
\begin{equation}\label{MR.Rr}
\Re\rho_\ka(\za)\geq\ba
\qa \ka\in\gK_\Ga.
\end{equation}
From
\hb{a_\ka\geq\ul{\al}} we infer that
\hb{|\rho_\ka(\za)|\leq c} for
\hb{\ka\in\gK_\Ga}. Set
\begin{equation}\label{MR.v}
v_\ka(\za)(s):=e^{-\rho_\ka(\za)s}
\qa s\geq0.
\npbd
\end{equation}
Then $\BC v_\ka(\za)$~is the subspace of $C_0\RC$ of decaying solutions
of~\eqref{MR.vR}.\po

\par
Let
\hb{\ka\in\gK_{\Ga_0}} so that
\hb{\cB_\ka=\ga}, the Dirichlet operator on~$\pl\BH^m$. Then
(recall~\eqref{E.v}),
\,\hb{R_\ka(\za)\eta=\eta v_\ka(\za)}. Thus, by~\eqref{MR.Rr},
$$
\|R_\ka(\za)\|_{\cL(\BC,C_0(\BR_+,\BC))}\leq 1
\qa \ka\in\gK_{\Ga_0}.
$$

\par
Now assume
\hb{\ka\in\gK_{\Ga_1}}. Then we see from \eqref{MR.B} and \eqref{MR.v} that
\begin{equation}\label{MR.B0}
\cB_\ka(\pl;\za)v_\ka(\za)(0)=-\imi a_\ka(x)\rho_\ka(\za).
\end{equation}
Consequently,
$$
\|R_\ka(\za)\|_{\cL(\BC,C_0(\BR_+,\BC))}
=\frac1{a_\ka(x)\,|\rho_\ka(\za)|}
\leq\frac1{\ul{\al}\ba}
\qa \ka\in\gK_{\Ga_1}.
$$
This proves that $(\cA_\ka,\cB_\ka)$ satisfies the uniform
parameter-dependent LS condition,
unif.\ w.r.t.\
\hb{\ka\in\gK_\Ga} and
\hb{t\in J}.

\begin{proposition}\label{pro-MR.2}
It holds
$$
\bigl(\pl_t+\cA_\ka,\,(\cB_\ka,\ga_0)\bigr)
\in\Lis\bigl(W_{\coW\ka}^{2/\mf2},
\,W_{\coW\ka}^{0/\mf2}
\oplus[\pl W_{\coW\ka}\oplus\ga_0W_{\coW\ka}]_{\cB_\ka}\bigr)
\qquad\text{$\gK_\Ga$-unif.}
$$
\end{proposition}

\begin{proof}
We deduce from \eqref{C.RB}, \eqref{C.0}, and
\cite[Example~VIII.1.8.6]{Ama19a} that
$$
[\pl W_{\coW\ka}\oplus\ga_0W_{\coW\ka}]_{\cB_\ka}
$$
is a well-defined closed linear subspace of
\hb{\pl W_{\coW\ka}\oplus\ga_0W_{\coW\ka}} and, using
also~\eqref{C.Aku},
$$
\bigl(\pl_t+\cA_\ka,\,(\cB_\ka,\ga_0)\bigr)
\in\cL\bigl(W_{\coW\ka}^{2/\mf2},
\,W_{\coW\ka}^{0/\mf2}
\oplus[\pl W_{\coW\ka}\oplus\ga_0W_{\coW\ka}]_{\cB_\ka}\bigr)
\qquad\text{$\gK_\Ga$-unif.}
\npbd
$$
The uniform LS~condition implies now the remaining assertions. For this we
refer to~\cite{AmaVolIII21a}.
\end{proof}
Nonhomogeneous linear parabolic boundary value problems (of arbitrary order
and in a Banach-space-valued setting) on Euclidean domains have been studied
in~\cite{DHP07a}. It follows, in particular from Proposition~6.4 therein,
that the isomorphism assertion is true for each
\hb{\ka\in\gK_\Ga}. However, it is not obvious whether the
\hbox{$\gK_\Ga$-uniformity} statement does also follow from the results
in~\cite{DHP07a}. For this one would have to check carefully the dependence
of all relevant estimates on the various parameters involved,
which would be no easy task. (The same observation applies to
Proposition~\ref{pro-MR.1}.) In~\cite{AmaVolIII21a} we present an
alternative proof which takes care of the needed uniform estimates.

\par
Lastly, we assume that
\hb{\ka\in\gK_S}. We set, once more suppressing a fixed
\hb{t\in J},
$$
a_\ka^1(x):=a_\ka(x)
\qb a_\ka^2(x):=a_\ka(-x)
\qa x\in\BH^m,
$$
and
$$
\gsa_\ka:=\diag[a_\ka^1,a_\ka^2]\sco\BH^m\ra\BC^{2\times 2}.
$$
Then
$$
\gA_\ka\gsu:=-\tdiv_\ka(\gsa_\ka\grad_\ka\gsu)
\qa \gsu=(u^1,u^2).
$$
Furthermore,
\hb{\gB_\ka=(\gB_\ka^0,\gB_\ka^1)}, where
\begin{equation}\label{MR.B2}
\bal
\gB_\ka^0\gsu
&:=\ga u^1-\ga u^2,\cr
\gB_\ka^1\gsu
&:=\bigl(\nu_\ka\bsn\ga(a_\ka^1\grad_\ka u^1+a_\ka^2\grad_\ka u^2)
 \bigr)_{g_\ka}
\eal
\npbd
\end{equation}
on~$\pl\BH^m$.\po

\par
Clearly,
$$
\sa\bigl(\gsa(x)\bigr)\is[\Re z\geq\ul{\al}]
\qa x\in\BH^m.
\npbd
$$
Thus $\gA_\ka$~is uniformly normally elliptic on~$\BH^m$,
\hbox{$\gK_S$-unif.}\po

\par
We define~$\rho_\ka^i$,
\,\hb{1=1,2}, by replacing~$a_\ka$ in \eqref{MR.rh} by~$a_\ka^i$ and
introduce~$v_\ka^i$ by changing~$\rho_\ka$ in\eqref{MR.v} to~$\rho_\ka^i$.
Then
$$
\BC v_\ka^1\oplus\BC v_\ka^2
$$
is the subspace of $C_0(\BR_+,\BC^2)$ of decaying solutions of
$$
\bigl(\lda+\gss\gA_\ka(x,\,\xi+\imi\nu_\ka(x)\pl)\bigr)\gsv=0
\qa x\in\pl\BH^m
\qb \gsv=(v^1,v^2).
$$
From \eqref{MR.B0} and \eqref{MR.B2} we see that the initial conditions
in~\eqref{E.LS} are in the present case
$$
\bal
v^1(0)-v^2(0)
&=\eta^1,\cr
a_\ka^1(x)\rho_\ka^1(x)
v^1(0)+a_\ka^2(x)\rho_\ka^2(x)v^2(0)
&=\imi\eta^2.
\eal
$$
Omitting
\hb{x\in\pl\BH^m}, the solution of this system is
$$
\bal
v_\ka^1(0)
&=v_\ka^2(0)+\eta^1,\cr
v_\ka^2(0)
&=\frac1{a_\ka^1\rho_\ka^1+a_\ka^2\rho_\ka^2}
 (\imi\eta^2-a_\ka^1\rho_\ka^1\eta^1).
\eal
$$
From this, the uniform boundedness of~$a_\ka$, and
\hb{\Re(a_\ka^1\rho_\ka^1+a_\ka^2\rho_\ka^2)\geq1/\ul{\al}\ba}
it follows that $(\gA_\ka,\gB_\ka)$ satisfies the uniform
parameter-dependent LS condition, unif.\ w.r.t.\
\hb{\ka\in\gK_S} and
\hb{t\in J}.

\begin{proposition}\label{pro-MR.3}
It holds
$$
\bigl(\pl_t+\cA_\ka,\ (\cC_\ka,\ga_0)\bigr)
\in\Lis\bigl(\bar W_{\coW\ka}^{2/\mf2},
\,\bar W_{\coW\ka}^{0/\mf2}
\oplus[\pl_S\bar W_{\coW\ka}\oplus\ga_0\bar W_{\coW\ka}]_{\cC_\ka}\bigr)
\npbd
$$
unif.\ w.r.t.\
\hb{\ka\in\gK_S} and
\hb{t\in J}.
\end{proposition}

\begin{proof}
Set
\hb{\gsu(x):=\bigl(u(x),u(-x)\bigr)} for
\hb{x\in\BH^m} and
\hb{\bar\gW_\ka^s:=\bar W_\ka^s\oplus\bar W_\ka^s} etc. Then the assertion
is true iff
$$
\bigl(\pl_t+\gA_\ka,\ (\gB_\ka,\ga_0)\bigr)
\in\Lis\bigl(\bar\gW_\ka^{2/\mf2},
\,\bar\gW_\ka^{0/\mf2}
\oplus[\pl_S\bar\gW_\ka\oplus\ga_0\bar\gW_\ka]_{\gB_\ka}\bigr)
\qquad\text{$\gK_S$-unif.}
$$
By the preceding considerations, the proof of Proposition~\ref{pro-MR.2}
applies verbatim to the system for~$\gsu$. This proves the claim.
\end{proof}

\section{Localizations}\label{sec-LT}
We presuppose \eqref{L.ass} and fix an \hbox{$S$-adapted} atlas for~$M$
with
\hb{\diam(U_{\coU\ka})<\ve/2} for
\hb{\ka\in\gK_S}. Then
$$
\gN(\ka):=\{\,\wk\in\gK\ ;\ U_{\coU\ka}\cap U_\wk\neq\es\,\}
\qa \ka\in\gK,
$$
\hb{\gN_\Ga(\ka):=\gN(\ka)\cap\gK_\Ga},
and \hb{\gN_S(\ka):=\gN(\ka)\cap\gK_S}. By the finite multiplicity of~$\gK$,
\begin{equation}\label{LT.k}
\card\bigl(\gN(\ka)\bigr)\leq c
\qa \ka\in\gK.
\end{equation}
We set for
\hb{\ka\in\gK} and
\hb{\wk\in\gN(\ka)}
$$
S_\kwk u:=\ka_*\wk^*u=u\circ(\wk\circ\ka^{-1})
\qa u\in\sW_\wk^{0/\mf2}.
$$
It follows from \eqref{U.K}(ii) that, given
\hb{s\in[0,2]},
\begin{equation}\label{LT.S}
S_\kwk\in\cL(\sW_\wk^{s/\mf2},\sW_{\coW\ka}^{s/\mf2})
\qquad\text{$\gK$-unif.}
\end{equation}

\par
Interpreting~$\gK$ as an index set, we put
\begin{equation}\label{LT.Wp}
\mfsW^{s/\mf2}:=\prod_{\ka\in\gK}\sW_{\coW\ka}^{s/\mf2},
\end{equation}
endowed with the product topology. For
\hb{\al\in\{0,\Ga\}} we set
$$
\mfW^{s/\mf2}[\al]:=\prod_{\ka\in\gK_\al}W_{\coW\ka}^{s/\mf2}
\qb \bar\mfW^{s/\mf2}:=\prod_{\ka\in\gK_S}\bar W_{\coW\ka}^{s/\mf2}.
$$
Since $\gK$~is the disjoint union of $\gK_0$, $\gK_\Ga$, and~$\gK_S$,
\begin{equation}\label{P.Ws}
\mfsW^{s/\mf2}
=\mfW^{s/\mf2}[0]\oplus\mfW^{s/\mf2}[\Ga]\oplus\bar\mfW^{s/\mf2}.
\end{equation}
A similar definition and direct sum decomposition applies
to $\ga_0\mfsW$. We also set
$$
\pl\mfW:=\prod_{\ka\in\gK_\Ga}\pl W_{\coW\ka}
\qb \pl_S\mfW:=\prod_{\ka\in\gK_S}\pl_S\bar W_{\coW\ka}.
$$

\par
We define linear operators $\cR$
and~$\cR^c$ by
\begin{equation}\label{LT.RR}
\cR\mfu:=\sum_\ka\pi_\ka\ka^*u_\ka
\qb \cR^cu:=\bigl(\ka_*(\pi_\ka u)\bigr)_{\ka\in\gK}
\end{equation}
for
\hb{\mfu=(u_\ka)\in\mfsW^{0/\mf2}} and
\hb{u\in L_1\bigl(J,L_{1,\loc}(M\ssm S)\bigr)}, respectively.
The sum is locally finite
and $\pi_\ka$~is identified with the multiplication operator
\hb{v\mt\pi_\ka v}.

\par
We want to evaluate
\hb{\cA\circ\cR\mfu} for
\hb{\mfu\in\mfsW^{2/\mf2}}. Observe
$$
\cA(\pi_\ka u)=\pi_\ka\cA u+[\cA,\pi_\ka]u
\qa u\in W_{\coW p}^{2/\mf2},
$$
the commutator being given by
\begin{equation}\label{LT.Ap}
[\cA,\pi_\ka]u
=-(\grad\pi_\ka\sn a\grad u)-\tdiv(au\grad\pi_\ka).
\end{equation}
Thus we get from \eqref{LT.RR}
\begin{equation}\label{LT.AR}
\cA\cR\mfu
=\sum_\ka\cA(\pi_\ka\ka^*u_\ka)
=\sum_\ka\pi_\ka\cA(\ka^*u_\ka)+\sum_\ka[\cA,\pi_\ka]\ka^*u_\ka.
\end{equation}
By
\hb{\cA(\ka^*u_\ka)=\ka^*\cA_\ka u_\ka}, the first sum
equals~$\cR\mfsA\mfu$, where
\hb{\mfsA:=\diag[\cA_\ka]}. Using
\hb{1=\sum_\wk\pi_\wk^2}, we find
\begin{equation}\label{LT.AR2}
\bal
{}
[\cA,\pi_\ka]\ka^*u_\ka
&=\sum_\wk\pi_\wk\pi_\wk[\cA,\pi_\ka]\ka^*u_\ka\cr
&=\sum_\wk\pi_\wk\wk^*
 \bigl((\wk_*\pi_\wk)\wk_*[\cA,\pi_\ka]\wk^*
 (\wk_*\ka^*)u_\ka\bigr)\cr
&=\sum_{\wk\in\gN(\ka)}\pi_\wk\wk^*
 \bigl((\wk_*\pi_\wk)[\cA_\wk,S_\wkk(\ka_*\pi_\ka)]S_\wkk u_\ka\bigr).
\eal
\end{equation}
Set
$$
\cA_\kwk:=(\ka_*\pi_\ka)
\bigl[\cA_\ka,S_\kwk(\wk_*\pi_\wk)\bigr]S_\kwk\chi.
$$
Then \eqref{LT.S}, \eqref{LT.Ap}, \eqref{M.LS}, \eqref{C.ak}, and
\hb{\ka_*\pi_\ka=(\ka_*\pi_\ka)\chi}
imply
\begin{equation}\label{LT.Akk}
\|\cA_\kwk v\|_{\sW_{\coW\ka}^{0/\mf2}}
\leq c\,\|\chi v\|_{\sW_{\coW\wk}^{1/\mf2}}
\qa \wk\in\gN(\ka)
\qb \ka\in\gK.
\end{equation}
We define
\hb{\mfsA_\ka^0\sco\mfsW^{1/\mf2}\ra\sW_{\coW\ka}^{0/\mf2}} by
$$
\mfsA_\ka^0\mfu:=\sum_{\wk\in\gN(\ka)}\cA_\kwk u_\wk
\qa \mfu\in\mfsW^{1/\mf2}
\qb \ka\in\gK.
$$
Then we deduce from \eqref{LT.k} and \eqref{LT.Akk} that
\begin{equation}\label{LT.Ak}
\|\mfsA_\ka^0\mfu\|_{\sW_{\coW\ka}^{0/\mf2}}
\leq c\sum_{\wk\in\gN(\ka)}\|\chi u_\wk\|_{\mfsW_\wk^{1/\mf2}}
\qa u\in\mfsW^{1/\mf2}
\qb \ka\in\gK.
\npbd
\end{equation}
Moreover,
\hb{\mfsA^0:=(\mfsA_\ka^0)_{\ka\in\gK}}.\po

\par
Now we sum \eqref{LT.AR2} over
\hb{\ka\in\gK} and interchange the order of summation to find
that the second sum in \eqref{LT.AR} equals~$\cR\mfsA^0\mfu$.
Thus, in total,
\begin{equation}\label{LT.RAA}
\cA\cR=\cR(\mfsA+\mfsA^0).
\end{equation}

\par
Similar considerations lead to
\begin{equation}\label{LT.AAR}
\cR^c\cA=(\mfsA+\wt{\mfsA}\vph{\sA}^0)\cR^c.
\end{equation}
Here
\hb{\wt{\mfsA}\vph{\sA}^0=(\wt{\mfsA}\vph{\sA}_\ka^0)_{\ka\in\gK}} with
\hb{\wt{\mfsA}\vph{\sA}_\ka^0\in\cL(\mfsW^{1/\mf2},\sW_{\coW\ka}^{0/\mf2})}
satisfying
\begin{equation}\label{LT.At}
\|\wt{\mfsA}\vph{\sA}_\ka^0\mfu\|_{\sW_{\coW\ka}^{0/\mf2}}
\leq c\sum_{\wk\in\gN(\ka)}\|\chi u_\wk\|_{\mfsW_\wk^{1/\mf2}}
\qa \mfu\in\mfsW^{1/\mf2}
\qb \ka\in\gK.
\end{equation}

\par
We turn to~$\Ga$ and define
\hb{\ka^\fdot:=\ga\ka} for
\hb{\ka\in\gK_\Ga}. Then
\begin{equation}\label{LT.Rga}
\cR_\Ga\mfu:=\sum_{\ka\in\gK_\Ga}\ga\pi_\ka(\ka^\fdot)^*u_\ka
\qa \mfu=(u_\ka)\in\pl\mfW,
\end{equation}
and
\begin{equation}\label{LT.Rgc}
\cR_\Ga^cu:=\bigl(\ka_*^\fdot\ga(\pi_\ka u)\bigr)_{\ka\in\gK_\Ga}
\qa u\in W_{\coW p}^{2/\mf2}\bigl((M\ssm S)\times J\bigr).
\end{equation}

\par
Observe that
$$
\cB(\pi_\ka u)=(\ga\pi_\ka)\cB u+[\cB,\pi_\ka]u
$$
and
$$
[\cB,\pi_\ka]u=\da\bigl(\nu\bsn\ga(a\grad\pi_\ka)\bigr)_{g^\ifdot}\ga u
\qa u\in W_{\coW p}^{2/\mf2}\bigl((M\ssm S)\times J\bigr).
$$
Similarly as above, we find
\begin{equation}\label{LT.RBB}
\cB\cR=\cR_\Ga(\mfB+\mfB^0),
\end{equation}
where
\hb{\mfB:=\diag[\cB_\ka]} and
\hb{\mfB^0=(\mfB_\ka^0)_{\ka\in\gK_\Ga}} with
\hb{\mfB_\ka^0\sco\mfW^{1/\mf2}[\Ga]\ra\{0\}\oplus\pl_1W_{\coW\ka}}
satisfying
\begin{equation}\label{LT.Bk}
\|\mfB_\ka^0\mfu\|_{\pl_1W_{\coW\ka}}
\leq c\sum_{\wk\in\gN_\Ga(\ka)}\|\chi u_\wk\|_{W_\wk^{1/\mf2}}
\qa \ka\in\gK_\Ga.
\end{equation}
Analogously,
\begin{equation}\label{LT.BBR}
\cR_\Ga^c\cB=(\mfB+\wt{\mfB}\vph{\mfB}^0)\cR^c,
\end{equation}
where
\hb{\wt{\mfB}\vph{\mfB}^0=(\wt{\mfB}\vph{\mfB}_\ka^0)} with
\hb{\wt{\mfB}\vph{\mfB}_\ka^0
   \sco\mfW^{1/\mf2}[\Ga]\ra\{0\}\oplus\pl_1W_{\coW\ka}} is such that
\begin{equation}\label{LT.Bk0}
\|\wt{\mfB}\vph{\mfB}_\ka^0\mfu\|_{\pl_1W_{\coW\ka}}
\leq c\sum_{\wk\in\gN_\Ga(\ka)}\|\chi u_\wk\|_{W_\wk^{1/\mf2}}
\qa \ka\in\gK_\Ga.
\end{equation}
Concerning the transmission interface~$S$, we define
$\cR_S$ and~$\cR_S^c$ analogously to \eqref{LT.Rga} and \eqref{LT.Rgc}.
Observe that
$$
[\cC,\pi_\ka]u
=\bigl(0,\eea{(\nu_S\sn a\grad\pi_\ka)u}\bigr).
$$
From this it is now clear that
\begin{equation}\label{LT.CCR}
\cC\cR=\cR_S(\mfC+\mfC^0)
\qb \cR_S^c\cC=(\mfC+\wt{\mfC}\vph{\mfC}^0)\cR^c,
\end{equation}
where
\hb{\mfC:=\diag[\cC_\ka]},
\ \hb{\mfC^0=(\mfC_\ka^0)}, and
\hb{\wt{\mfC}\vph{\mfC}^0=(\wt{\mfC}\vph{\mfC}_\ka^0)} with
\begin{equation}\label{LT.Ck}
\bal
\|C_\ka^0\mfu\|_{\pl_1W_{\coW\ka}}
&\leq c\sum_{\wk\in\gN_S(\ka)}\|\chi u_\wk\|_{\bar W_\wk^{1/\mf2}},\cr
\|\wt{C}\vph{C}_\ka^0\mfu\|_{\pl_1W_{\coW\ka}}
&\leq c\sum_{\wk\in\gN_S(\ka)}\|\chi u_\wk\|_{\bar W_\wk^{1/\mf2}}
\eal
\npbd
\end{equation}
for
\hb{\ka\in\gK_S} and
\hb{\mfu\in\bar\mfW^{2/\mf2}}.\po

\par
The following consequences of the preceding results are needed
to establish Theorem~\ref{thm-L.MR}.

\begin{lemma}\label{lem-LT.ch}
Fix
\hb{s\in(1,2)} and
\hb{q>p} with
\hb{1/p-1/q<(2-s)/(m+2)}.\break
Then
$$
(v\mt\chi v)
\in\cL\bigl(\sW_{\coW\ka}^{2/\mf2},L_q(J,\sW_{\coW\ka}^s)
\cap W_{\coW q}^{s/2}(J,\sW_{\coW\ka}^0)\bigr)
\qquad\text{$\gK$-unif.}
$$
\end{lemma}

\begin{proof}
(1)
Set
\hb{\BX:=\BR^m} and
\hb{s_1:=s+(m+2)(1/p-1/q)<2}. By (VII.3.6.3) and Example~VII.3.6.5
of~\cite{Ama19a}
\begin{equation}\label{LT.Ws2}
W_{\coW q}^{s/\mf2}
\doteq L_q(J,W_{\coW q}^s)\cap W_{\coW q}^{s/2}(J,L_q),
\end{equation}
where
\hb{W_{\coW q}^{s/\mf2}=W_{\coW q}^{s/\mf2}(\BX\times J)} etc. Hence, see
\cite[Theorem VII.2.2.4(iv)]{Ama19a},
$$
W_{\coW p}^{2/\mf2}\hr W_{\coW p}^{s_1/\mf2}
\hr L_q(J,W_{\coW q}^s)\cap W_{\coW q}^{s/2}(J,L_q).
$$
Consequently, invoking \eqref{C.W},
$$
W_{\coW\ka}^{2/\mf2}
\hr L_q(J,W_{\coW q,\ka}^s)\cap W_{\coW q}^{s/2}(J,L_{q,\ka})
\qquad\text{$\gK_0$-unif.}
$$

\par
(2)
Since
\hb{\supp(\chi)\is(-1,1)^m}, H\"older's inequality implies
$$
\|\chi v\|_{W_{\coW p}^k}\leq c\,\|\chi v\|_{W_{\coW q}^k}
\qa v\in W_{\coW q}^2
\qb k=0,1,2.
$$
Hence, by interpolating and then using \eqref{C.W} once more,
$$
\|\chi v\|_{W_{\coW\ka}^\sa}\leq c\,\|\chi v\|_{W_{\coW q,\ka}^\sa}
\qquad\text{$\gK_0$-unif.}
\qb \sa\in\{0,s\}.
$$
From this we get
$$
\|\chi u\|_{L_q(J,W_{\coW\ka}^s)}
\leq c\,\|\chi u\|_{L_q(J,W_{\coW q,\ka}^s)}
\qquad\text{$\gK_0$-unif.}
$$
and
$$
\|\chi u\|_{W_{\coW q}^{s/2}(J,W_{\coW\ka}^0)}
\leq c\,\|\chi u\|_{W_{\coW q}^{s/2}(J,W_{\coW q,\ka}^0)}
\qquad\text{$\gK_0$-unif.}
\npbd
$$
Now the assertion follows in this case from step~(1).

\par
(3)
The preceding arguments also hold if we replace
\hb{\BX=\BR^m} by
\hb{\BX=\BH^m}. Then (see Remark~\ref{rem-F.H}) it also applies to
\hb{\BX=\BR^m\ssm\pl\BH^m}. Thus the lemma is proved.
\end{proof}
Given a function space~$\gF$ defined on~$J$, we write~$\gF(\tau)$ for its
restriction to~$J_\tau$,
\ \hb{0<\tau\leq T}.

\begin{lemma}\label{lem-LT.A0}
Let
\hb{\wh{\mfsA}_\ka\in\{\mfsA_\ka^0,\wt{\mfsA}_\ka\}}. There exists
\hb{\ve>0} such that
$$
\|\wh{\mfsA}_\ka\mfu\|_{\sW_{\coW\ka}^{0/\mf2}(\tau)}
\leq c\coc\tau^\ve\sum_{\wk\in\gN(\ka)}\|u_\wk\|_{\sW_\ka^{2/\mf2}(\tau)}
\qa \mfu\in\mfsW^{2/\mf2},
\npbd
$$
unif.\ w.r.t.\
\hb{\ka\in\gK} and
\hb{0<\tau\leq T}.
\end{lemma}

\begin{proof}
Fix $s$ and~$q$ as in Lemma~\ref{lem-LT.ch} and set
\hb{\ve:=1/p-1/q}. We get from H\"older's inequality, Lemma~\ref{lem-LT.ch},
\eqref{LT.Ak}, and \eqref{LT.At}
$$
\bal
\|\wh{\mfsA}_\ka\mfu\|_{\sW_{\coW\ka}^{0/\mf2}(\tau)}
&=\|\wh{\mfsA}_\ka\mfu\|_{L_p(J_\tau,\sW_{\coW\ka}^0)}
 \leq\tau^\ve\,\|\wh{\mfsA}_\ka\mfu\|_{L_q(J_\tau,\sW_{\coW\ka}^0)}\cr
&\leq c\coc\tau^\ve\sum_{\wk\in\gN(\ka)}
 \|\chi u_\wk\|_{L_q\cap\sW_\wk^{s/2}(\tau)}
 \leq c\coc\tau^\ve\sum_{\wk\in\gN(\ka)}\|u_\wk\|_{\sW_\ka^{2/\mf2}(\tau)}
\eal
$$
for
\hb{0<\tau\leq T}, \hbox{$\gK$-unif.}, where
\hb{\Vsdot_{L_q\cap\sW_{\coW\ka}^{s/2}}} is the norm in the image space
occurring in Lemma~\ref{lem-LT.ch}.
\end{proof}
The next lemma provides analogous estimates for the boundary and
transmission operators.

\begin{lemma}\label{lem-LT.BC}
Let
\hb{\wh{\mfB}_\ka\in\{\mfB_\ka^0,\wt{\mfB}\vph{\mfB}_\ka^0\}} and
\hb{\wh{\mfC}_\ka\in\{\mfC_\ka^0,\wt{\mfC}\vph{\mfC}_\ka^0\}}. There exists
\hb{\ve>0} such that
$$
\|\wh{\mfB}_\ka\mfu\|_{\pl W_{\coW\ka}(\tau)}
\leq c\coc\tau^\ve\sum_{\wk\in\gN_\Ga(\ka)}
\|u_\wk\|_{W_{\coW\wk}^{2/\mf2}(\tau)}
\qquad\text{$\gK_\Ga$-unif.}
\qb \mfu\in\mfW^{2/\mf2}[\Ga],
$$
and
$$
\|\wh{\mfC}_\ka\mfu\|_{\pl_SW_{\coW\ka}(\tau)}
\leq c\coc\tau^\ve\sum_{\wk\in\gN_S(\ka)}
\|u_\wk\|_{\bar W_{\coW\wk}^{2/\mf2}(\tau)}
\qquad\text{$\gK_S$-unif.}
\qb \mfu\in\bar\mfW^{2/\mf2},
\npbd
$$
for
\hb{0<\tau\leq T}.
\end{lemma}\po

\begin{proof}
Let $s$, $q$, and~$\ve$ be as in the preceding proof.

\par
Given any Banach space~$E$, H\"older's inequality gives
$$
\|u\|_{W_{\coW p}^k(J_\tau,E)}
\leq\tau^\ve\,\|u\|_{W_{\coW q}^k(J_\tau,E)}
\qa k=0,1.
$$
Hence, by interpolation (see \cite[Theorems VII.2.7.4
and VII.7.3.4]{Ama19a}),
\begin{equation}\label{LT.Wpp}
\|u\|_{W_{\coW p}^{(1-1/p)/2}(J_\tau,E)}
\leq c\coc\tau^\ve\,\|u\|_{W_{\coW q}^{(1-1/p)/2}(J_\tau,E)}
\qa 0<\tau<T.
\end{equation}

\par
Set
$$
\cL_\ka^\sa(\tau)
:=L_q\bigl(J_\tau,W_{\coW\ka}^\sa(\pl\BH^m)\bigr)
\cap W_{\coW q}^{\sa/2}\bigl(J_\tau,W_{\coW\ka}^0(\pl\BH^m)\bigr)
\qa \ka\in\gK_\Ga.
$$
Then we get from \eqref{LT.Wpp}
\begin{equation}\label{LT.L}
\Vsdot_{\pl_1W_{\coW\ka}(\tau)}
\leq c\coc\tau^\ve\,\Vsdot_{\cL_\ka^{1-1/p}(\tau)}
\qquad\text{$\gK_\Ga$-unif.}
\qb 0<\tau<T.
\end{equation}
It is clear from the structure of~$\wh{\mfB}_\ka$ and the mapping
properties of~$\ga$ that
$$
\|\wh\mfB_\ka\mfu\|_{\cL_\ka^{1-1/p}(\tau)}
\leq c\,\|\chi\mfu\|_{\cL_\ka^1(\tau)}
\qquad\text{$\gK_\Ga$-unif.}
$$
Since
\hb{\cL_\ka^s(\tau)\hr\cL_\ka^1(\tau)} \hbox{$\gK_\Ga$-unif.} and unif.\
w.r.t.~$\tau$, the first assertion now follows from \eqref{LT.L},
Lemma~\ref{lem-LT.ch}, and the fact that $\wh\mfB_\ka$~has its image in the
closed  linear subspace $\pl_1W_{\coW\ka}(\tau)$
of~$\pl W_{\coW\ka}(\tau)$.

\par
The proof of the second claim is similar.
\end{proof}\po

\section{Proof of Theorem~\ref{thm-L.MR}}\label{sec-P}
Let
\hb{\mfE=\prod_{\al\in\sA}E_\al}, where each~$E_\al$ is a Banach space
and $\sA$~is a countable index set. Then
$\ell_p(\mfE)$ is the Banach space of \hbox{$p$-summable}
sequences in~$\mfE$.

\par
We put
$$
\BE^i[\al]:=\ell_p\bigl(\mfW^{i/\mf2}[\al]\bigr)
\qb \ga_0\BE[\al]:=\ell_p\bigl(\ga_0\mfW[\al]\bigr)
\qa \al\in\{0,\Ga\},
$$
and
$$
\bar\BE^i:=\ell_p\bigl(\bar\mfW^{i/\mf2}\bigr)
\qb \ga_0\bar\BE:=\ell_p(\ga_0\bar\mfW)
$$
for
\hb{i=0,2}. Then
$$
\BE^i:=\BE^i[0]\oplus\BE^i[\Ga]\oplus\bar\BE^i
\qb i=0,2
\qa \ga_0\BE:=\ga_0\BE[0]\oplus\ga_0\BE[\Ga]\oplus\ga_0\bar\BE.
\npbd
$$
Moreover,
\hb{\BF_\Ga:=\ell_p(\pl\mfW)},
\ \hb{\BF_S:=\ell_p(\pl_S\bar\mfW)}.

\par
Recall the definitions of $\cRcRc$, $(\cR_\Ga,\cR_\Ga^c)$, and
$(\cR_S,\cR_S^c)$ in Section~\ref{sec-LT}.

\setcounter{proposition}{0}
\begin{proposition}\label{pro-P.R}
$\cRcRc$ is an \hbox{r-c} pair for $(\BE^k,\bar W^{k/\mf2})$,
\,\hb{k=0,2}, and
$$
(\cR_\Ga\oplus\cR_S\oplus\cR,\,\cR_\Ga^c\oplus\cR_S^c\oplus\cR^c)
$$
is an \hbox{r-c} pair for
$$
(\BF_\Ga\oplus\BF_S\oplus\ga_0\bar\BE,\,\pl W\oplus\pl_SW\oplus\ga_0\bar W).
$$
\end{proposition}

\begin{proof}
This follows from \cite[Theorem~6.1]{Ama12b} (also see
\cite[Theorem~9.3]{Ama12c} or \cite{AmaVolIII21a}).
\end{proof}\po

\setcounter{lemma}{1}
\begin{lemma}\label{lem-P.Gr}
$\ga_0$~is a retraction from~$\bar W^{2/\mf2}$ onto~$\ga_0\bar W$ and
from~$\BE^2$ onto~$\ga_0\bar\BE$. Furthermore,
\hb{\ga_0\cR=\cR\ga_0}.
\end{lemma}

\begin{proof}
Due to \eqref{C.0} there exists
\hb{\ga_0^c\in\cL(\ga_0\bar\BE,\BE^2)} such that $(\ga_0,\ga_0^c)$ is
an \hbox{r-c} pair for~$(\BE^2,\ga_0\bar\BE)$. For the moment we denote it
by~$(\bar\ga_0,\bar\ga_0^c)$. Then it follows from
Proposition~\ref{pro-P.R} that
$$
\cR\bar\ga_0\cR^c\in\cL(\bar W^{2/\mf2},\ga_0\bar W)
\qb \cR\bar\ga_0^c\cR^c\in\cL(\ga_0\bar W,\bar W^{2/\mf2}).
$$
Since $\ga_0$~is the evaluation at
\hb{t=0} and $\cRcRc$~is independent of
\hb{t\in J}, we see that
\hb{\ga_0\cR=\cR\bar\ga_0}. Hence
\hb{\ga_0\in\cL(\bar W^{2/\mf2},\ga_0\bar W)} and
\hb{\ga_0^c:=\cR\bar\ga_0^c\cR^c} is a continuous right
inverse for~$\ga$.
\end{proof}

\par
We write
$$
\wh{\BA}:=\mfsA+\wh{\mfsA}
\text{ with }
\wh{\mfsA}:=[\wh{\mfsA}_\ka]_{\ka\in\gK}
$$
and, using \eqref{P.Ws}, set
\hb{\mfu=(\mfv,\mfw,\mfz)\in\mfsW^{2/\mf2}} and, analogously,
$$
\wh{\BB}\mfu:=(\mfB+\wh{\mfB})\mfw\in\pl\mfW
\qb \wh{\BC}\mfu:=(\mfC+\wh{\mfC})\mfz\in\pl_S\mfW.
\npbd
$$
Moreover,
\hb{\BG:=\BF_\Ga\oplus\BF_S\oplus\ga_0\bar\BE} and
$[\BG]_{\mfB,\mfC}$~is the  linear subspace  consisting of all
\hb{(\mfvp,\mf\psi,\mfu_0)} satisfying the compatibility conditions
$$
\mfB(0)\mfu_0 =\mfvp(0)
\qb \mfC(0)\mfu_0 =\mfpsi(0)
$$
if
\hb{p>3}, with the corresponding modifications if
\hb{p<3}. Analogous definitions apply to~$[\BG]_{\wh{\BB},\wh{\BC}}$.

\setcounter{proposition}{2}
\begin{proposition}\label{pro-P.is}
${}$
\hb{\bigl(\pl_t+\wh{\BA},\,(\wh{\BB},\wh{\BC},\ga_0)\bigr)
   \in\Lis\bigl(\BE^2,\,\BE^0\oplus[\BG]_{\wh{\BB},\wh{\BC}}\bigr)}.
\end{proposition}

\begin{proof}
(1)
First we prove that
\begin{equation}\label{P.D}
\BL:=\bigl(\pl_t+\mfsA,\,(\mfB,\mfC,\ga_0)\bigr)
\in\Lis\bigl(\BE^2,\,\BE^0\oplus[\BG]_{\mfB,\mfC}\bigr).
\end{equation}
Since $\BL$~has diagonal structure, the claim is a direct consequence of
Propositions \hbox{\ref{pro-MR.1}--\ref{pro-MR.3}}.

\par
(2)
Set
\hb{\BL_0:=(\wh{\mfsA},\wh{\mfB},\wh{\mfC},0)}. Then
\begin{equation}\label{P.DD}
\wh{\BL}:=
(\pl_t+\wh{\BA},\ \wh{\BB},\wh{\BC},\ga_0)=\BL+\wh{\BL}_0.
\end{equation}
It follows from \eqref{LT.k} and
Lemmas \ref{lem-LT.A0} and \ref{lem-LT.BC} that
\hb{\wh{\BL}_0\in\cL(\BE^2,\,\BE^0\oplus\BG)} and there exists
\hb{\ve>0} such that
\begin{equation}\label{P.t}
\|\wh{\BL}_0\|_{\cL(\BE^2(\tau),\,\BE^0(\tau)\oplus\BG(\tau))}
\leq c\coc\tau^\ve
\qa 0<\tau\leq T.
\end{equation}
From this and step~(1) we see that
\begin{equation}\label{P.L}
\wh{\BL}\in\cL(\BE^2,\,\BE^0\oplus\BG).
\end{equation}

\par
Write
$$
\BE_0^2:=\{\,\mfv\in\BE^2\ ;\ \ga_0\mfv=\mf0\,\}
\qb \BG_0:=\BF_\Ga\oplus\BF_S\oplus\{0\}.
$$
By Lemma~\ref{lem-P.Gr} we can choose a coretraction~$\ga_0^c$ for
\hb{\ga_0\in\cL(\BE^2,\ga_0\bar\BE)}. Let
\hb{(\mff,\mfg)\in\BE^0\oplus[\BG]_{\wh{\BB},\wh{\BC}}} with
\hb{\mfg=(\mfvp,\mfpsi,\mfu_0)}. Set
\hb{\ol{\mfu}:=\ga_0^c\mfu_0}. Then
\hb{\mfu\in\BE^2} satisfies
\hb{\wh{\BL}\mfu=(\mff,\mfg)} iff
\hb{\mfv:=\mfu-\ol{\mfu}} is such that
$$
\wh{\BL}\mfv=(\mff,\mfg)-\wh{\BL}\ol{\mfu}
=:(\mff_0,\mfg_0).
$$
Note that
\hb{\mfv\in\BE_0^2} and
\hb{\mfg_0\in[\BG_0]_{\wh{\BB},\wh{\BC}}}. Suppose
\hb{p>3}. Given any
\hb{\mfw\in\BE_0^2}, we get
$$
\ga_0(\mfB\mfw)=\mfB(0)\ga_0\mfw=0=\wh{\BB}(0)\ga_0\mfw=\ga_0(\wh{\BB}\mfw).
$$
From this, the analogous relation for $\mfC$ and~$\wh{\BC}$,
and \eqref{P.L} we infer that it suffices to prove that
\hb{\wh{\BL}\sco\BE_0^2\ra\BE^0\oplus[\BG_0]_{\mfB,\mfC}} is surjective
and has a continuous inverse. Obvious modifications apply to
\hb{p<3}.

\par
(3)
Let
\hb{\BF:=\BE^0\oplus[\BG_0]_{\mfB,\mfC}} and
\hb{\mfh\in\BF}. Suppose
\hb{\mfu\in\BE_0^2} and set
\hb{\mfv:=\BL\mfu\in\BF}. By step~(1) and \eqref{P.DD}, the equation
\hb{\wh{\BL}u=\mfh} is equivalent to
\hb{\mfv+\wh{\BL}_0\BL^{-1}\mfv=\mfh}. Observe
\hb{\wh{\BL}_0\BL^{-1}\mfv\in\BF}.

\par
Due to \eqref{P.t}, we can fix
\hb{\ol{\tau}\in(0,T]} such that
\hb{\|\wh{\BL}_0\BL^{-1}\|_{\cL(\BF(\tau))}\leq1/2}. As is well-known
(e.g.,~\cite[Lemma~12.2]{Ama17a}), this implies that
\hb{\wh{\BL}\in\Lis\bigl(\BF(\ol{\tau}),\BF(\ol{\tau})\bigr)}.

\par
(4)
If
\hb{\ol{\tau}=T}, then we are done. Otherwise we repeat this argument
for the problem in
\hb{[0,\,T-\ol{\tau}]} obtained by the time-shift
\hb{t\mt t-\ol{\tau}} and with the initial value~$\mfu(\ol{\tau})$. After
finitely many such steps we reach~$T$. The proposition is proved.
\end{proof}

\begin{proof}
{\it of Theorem~\ref{thm-L.MR}}
Now we write $(\BA,\BB,\BC)$ for $(\wh{\BA},\wh{\BB},\wh{\BC})$
if
$(\wh{\mfsA},\wh{\mfB},\wh{\mfC})$ equals $(\mfsA^0,\mfB^0,\mfC^0)$, and
$(\wt{\BA},\wt{\BB},\wt{\BC})$ otherwise. We also set
\hb{G:=\pl W\oplus\pl_SW\oplus\ga_0\bar W}. Then Proposition~\ref{pro-P.R}
implies
$$
\vecTeX\cR:=\cR\oplus(\cR_\Ga\oplus\cR_S\oplus\cR)
\in\cL(\BE^k\oplus\BG,\,\bar W^{k/\mf2}\oplus G)
$$
and
$$
\ {}
\vecTeX\cR^c:=\cR^c\oplus(\cR_\Ga^c\oplus\cR_S^c\oplus\cR^c)
\in\cL(\bar W^{k/\mf2}\oplus G,\,\BE^k\oplus\BG)
\npbd
$$
for
\hb{k=0,2}.\po

\par
Let
\hb{\bigl(\mfu,(\mfvp,\mfpsi,\mfu_0)\bigr)\in\BE^2\oplus\BG} and write
\hb{\bigl(u,(\vp,\psi,u_0)\bigr)}
for its image under~$\vecTeX\cR$. Then we obtain from \eqref{LT.RBB} and
\eqref{LT.CCR}
$$
\cB u=\cB\cR\mfu=\cR_\Ga\BB\mfu
\qb \cC u=\cC\cR\mfu=\cR_S\BC\mfu.
$$
Suppose
\hb{p>3},
\,\hb{\mfu_0=\ga_0\mfu}, and
\hb{\BB(0)\mfu_0=\mfvp(0)}. Then
$$
\bal
\vp(0)
&=\cR_\Ga\mfvp(0)
 =\cR_\Ga\BB(0)\mfu_0
 =\cR_\Ga\ga_0(\BB\mfu)\cr
&=\ga_0\cR_\Ga\BB\mfu
 =\ga_0\cB\cR\mfu=\ga_0(\cB u)
 =\cB(0)(\ga_0u).
\eal
$$

\par
Since Lemma~\ref{lem-P.Gr} and
\hb{\mfu_0=\ga_0\mfu} imply
\hb{u_0=\ga_0u}, we see that
\hb{\cB(0)u_0=\vp_0}. Similarly, we find that it follows from
\hb{\BC(0)\mfu_0=\mfpsi(0)} that
\hb{\cC(0)u_0=\psi(0)}. Thus, letting
\hb{\BF:=[\BG]_{\BB,\BC}} and
\hb{F:=[G]_{\cB,\cC}},
we have shown that
$$
\vecTeX\cR(\BE^2\oplus\BF)\is\bar W^{2/\mf2}\oplus F.
$$
Consequently,
\begin{equation}\label{P.R}
\vecTeX\cR\in\cL(\BE^2\oplus\BF,\,\bar W^{2/\mf2}\oplus F).
\end{equation}
We find analogously that
\begin{equation}\label{P.Rc}
\vecTeX\cR^c\in\cL(\bar W^{2/\mf2}\oplus F,\,\BE^2\oplus\BF).
\npbd
\end{equation}
This holds for
\hb{p>3}. The case
\hb{p<3} is similar.\po

\par
Now we set
$$
L:=\bigl(\pl_t+\cA,\,(\cB,\cC,\ga_0)\bigr)
\qb \BL:=\bigl(\pl+\BA,\,(\BB,\BC,\ga_0)\bigr).
$$
Define
\hb{\wt{\BL}:=\bigl(\pl+\wt{\BA},\,(\wt{\BB},\wt{\BC},\ga_0)\bigr)}.
It is a consequence
of \eqref{LT.RAA}, \eqref{LT.AAR}, \eqref{LT.RBB}, \eqref{LT.BBR},
\eqref{LT.CCR}, and the fact that $\cRcRc$ is independent of~$t$ that
\begin{equation}\label{P.LR}
L\cR=\vecTeX\cR\BL
\qb \vecTeX\cR^cL=\wt{\BL}\vecTeX\cR^c.
\end{equation}
Proposition~\ref{pro-P.is} guarantees
\begin{equation}\label{P.LL}
\BL,\wt{\BL}\in\Lis(\BE^2,\,\BE^0\oplus\BF).
\end{equation}
Suppose
\hb{Lu=0}. Then \eqref{P.LR} and \eqref{P.LL} imply
\hb{\cR^cu=0}. Thus
\hb{u=\cR\cR^cu=0}. This shows that $L$~is injective.

\par
Let
\hb{(f,g)\in\bar W^{0/\mf2}\oplus F}. By \eqref{P.LL} we find
\hb{\mfu\in\BE^2} satisfying
\hb{\BL\mfu=\vecTeX\cR^c(f,g)}. Put
$$
u:=\cR\mfu=\cR\BL^{-1}\vecTeX\cR^c(f,g)\in\bar W^{2/\mf2}.
$$
Then, by \eqref{P.LR} and \eqref{P.LL},
$$
Lu=L\cR\BL^{-1}\vecTeX\cR^c(f,g)
=\vecTeX\cR\BL\BL^{-1}\vecTeX\cR^c(f,g)
=\vecTeX\cR\vecTeX\cR^c(f,g)
=(f,g).
$$
Hence $L$~is surjective and, by \eqref{P.R} and \eqref{P.Rc},
$$
L^{-1}=\cR\BL^{-1}\vecTeX\cR^c
\in\cL(\bar W^{0/\mf2}\oplus F,\,\bar W^{2/\mf2}).
\npbd
$$
The proof is accomplished.
\end{proof}

\setcounter{remarks}{3}
\begin{remarks}\label{rem-P.P}
(a)
Recall that either some of $\Ga_0$, $\Ga_1$, and~$S$, or all of them,
can be empty. If
\hb{(\Ga,S)\neq\{\es,\es\}}, then the result is new. Otherwise, it is
a special case of the more general Theorem~1.23(ii) of~\cite{Ama17a}.

\par
(b)
Theorem~\ref{thm-L.MR} is true for systems
\hb{u=(u^1,\ldots,u^n)}, provided the uniform
Lopatinskii-Shapiro conditions apply. This is trivially the case if
$a$~is a diagonal matrix.\qed
\end{remarks}

\section{Membranes with Boundary}\label{sec-MB}
Now we turn to the case of membranes intersecting~$\Ga$ transversally. This
case is handled by reducing it to the situation studied in the preceding
section.

\begin{theorem}\label{thm-MB.P}
Let \eqref{S.ass} be satisfied. Theorem~\ref{thm-L.MR} applies with $\Mg$
replaced by $\whMwhg$.
\end{theorem}

\begin{proof}
Theorem~\ref{thm-S.M}.
\end{proof}\po
In preparation for the proof of Theorem~\ref{thm-R.RD} we derive rather
explicit representations of $(\cA,\cB,\cC)$ and the relevant function
spaces in a tubular neighborhood of
\hb{\Sa=\pl S} in~$\wh{M}$.

\par
We use the notations of Sections \ref{sec-H} and~\ref{sec-S} and set
$$
U:=U(\ve)
\qa \dot N:=N(\ve)\big\backslash\bigl\{(0,0)\bigr\}
\qb \wt{g}:=g_2/\rho^2.
$$
The Christoffel symbols~$\wt{\Ga}_{ij}^k$ for the metric~$\wt{g}$
turn out to be
$$
\wt{\Ga}_{11}^1=\wt{\Ga}_{12}^2=-\wt{\Ga}_{22}^1=-\rho^{-1}\pl_1\rho
\qb \wt{\Ga}_{22}^2=\wt{\Ga}_{12}^1=-\wt{\Ga}_{11}^2=-\rho^{-1}\pl_2\rho.
$$
We set
\hb{D:=(D_1,D_2)} with
\hb{D_i:=\rho\pl_i} and
\hb{\wt{\na}:=\na_{\wt{g}}}. Then, for
\hb{u\in\bar C^2(\dot N\ssm G_\sa)} with
\hb{G_\sa:=\graph(f_\sa)},
\begin{equation}\label{MB.N}
\bal
\rho^2\wt{\na}_{11}u
&=D_1^2u-\pl_2\rho D_2u,
    &\quad \rho^2\wt{\na}_{12}u
    &=D_1D_2u+\pl_2\rho D_1u,\cr
\rho^2\wt{\na}_{21}u
&=D_2D_1u+\pl_1\rho D_2u,
    &\quad \rho^2\wt{\na}_{22}u
    &=D_2^2u-\pl_1\rho D_1u.
\eal
\end{equation}
Hence, see \eqref{F.1} and \eqref{F.2},
\begin{equation}\label{MB.D1}
|\wt{\na}u|_{\wt{g}^*}^2
=|Du|^2=|D_1u|^2+|D_2u|^2
\end{equation}
and
\begin{equation}\label{MB.D2}
|\wt{\na}^2u|_{\wt{g}_0^2}^2
=\rho^4\Bigl((\wt{\na}_{11}u)^2+(\wt{\na}_{12}u)^2
 +(\wt{\na}_{21}u)^2+(\wt{\na}_{22}u)^2\Bigr).
\end{equation}

\par
Let
$$
[D^2u]^2:=(D_1^2u)^2+(D_1D_2u)^2+(D_2D_1u)^2+(D_2^2u)^2
$$
and
$$
\dl u\dr_\rho^2
:=|u|^2+|Du|^2+[D^2u]^2+|\na_hu|_{h^*}^2+|\na_h^2u|_{h_0^2}^2.
$$
We write
$$
\bar\cW_{\coW p}^2
=\bigl(\bar\cW_{\coW p}^2,\Vsdot_{\bar\cW_{\coW p}^2}\bigr)
$$
for the space of all
\hb{u\in L_{1,\loc}(\dot N\times\Sa)} for which the norm
$$
\|u\|_{\bar\cW_{\coW p}^2}
:=\Bigl(\int_\Sa\int_{\dot N\setminus G(\sa)}
\dl u\dr_\rho^p(x,y,\sa)\,\frac{d(x,y)}{\rho^2(x,y)}
\,d\vol_\Sa(\sa)\Bigr)^{1/p}
$$
is finite. Moreover, $\bar\cW_{\coW p}^0$~is obtained by replacing~
\hb{\dl\cdot\dr_\rho} by~
\hb{\vsdot}. It is then clear how to define the anisotropic
spaces~$\bar\cW_{\coW p}^{k/\mf2}$,
\,\hb{k=0,2}.

\setcounter{proposition}{1}
\begin{proposition}\label{pro-MB.N}
${}$
\hb{u\in\bar W_{\coW p}^2(U\ssm S;\wh{g})} iff
\hb{\chi_*u\in\bar\cW_{\coW p}^2}.
\end{proposition}

\begin{proof}
First we note that
\begin{equation}\label{MB.W}
\sqrt{\wt{g}}=\rho^{-2}.
\end{equation}
It follows from \eqref{MB.D2} that
$$
|\wt{\na}^2v|_{\wt{g}_0^2}^2
\leq c\bigl(|Dv|^2+[D^2v]^2\bigr).
$$
Consequently, since
\hb{\na=\chi^*(\wt{\na}\oplus\naSa)\chi_*},
\begin{equation}\label{MB.u}
\|u\|_{\bar W_{\coW p}^2(U\setminus S,\,\wh{g})}
\leq c\,\|\chi_*u\|_{\bar\cW_p^2}.
\end{equation}
If $(u_j)$~is a converging sequence in
\hb{\bar W_{\coW p}^2:=\bar W_{\coW p}^2(U\ssm S;\wh{g})},
then we infer from \hbox{\eqref{MB.N}--\eqref{MB.D2}}

that
$(\chi_*u_j)$ converges in~$\bar\cW_{\coW p}^2$. Hence
\hb{\chi_*u\in\bar\cW_{\coW p}^2} if
\hb{u\in\bar W_{\coW p}^2}. From this, \eqref{MB.u}, and
Banach's homomorphism theorem we obtain
$$
\|\chi_*\!\cdot\|_{\bar\cW_{\coW p}^2}
\sim\Vsdot_{\bar W_{\coW p}^2}.
\npbd
$$
This proves the claim.
\end{proof}
Let
\hb{\lda\sco V_\lda\ra(-1,1)^{m-2}},
\ \hb{\sa\mt z} be a local chart for~$\Sa$. Then
$$
\ka:=(\id_{\dot N}\times\lda)\circ\chi\sco U_{\coU\ka}\ra Q_\ka^m
\qb q\mt(x,y,z)=(x^1,x^2,\ldots,x^m)
$$
is a local chart for~$\wh{M}$ with
\hb{U_{\coU\ka}\is U} and
\hb{\ka(U_{\coU\ka})=\dot N\times(-1,1)^{m-2}}. Set
\hb{\wt{h}:=\lda_*h}. It follows from \eqref{S.g} that
\hb{\ka_*\wh{g}=\wt{g}+\wt{h}}.

\par
Assume
\hb{u\in C^2(\wh{M})}, put
\hb{v:=\ka_*u\in C^2(Q_\ka^m)}, and denote by
\hb{e_1,\ldots,e_m} the standard basis of~$\BR^m$. Then
\begin{equation}\label{MB.gr}
\ka_*(\grad_{\wh{g}}u)
=\grad_{\wt{g}}v+\grad_{\wt{h}}v
=\rho^2(\pl_x ve_1+\pl_yve_2)+\grad_{\wt{h}}v.
\end{equation}
Similarly, let
\hb{X=X^i\pl/\pl x^i\in C^1(TU_{\coU\ka})} and set
$$
Y:=\ka_*X=Y^ie_i=Y^1e_1+Y^2e_2+\wt{Y},
$$
where
\hb{\wt{Y}=Y^\al e_\al} with $\al$~running from~$3$ to~$m$. Observe that $v$
and~$Y$ depend on $(x,y,z)$. It follows
$$
\ka_*(\tdiv_{\wh{g}}X)
=\ka_*\Bigl(\frac1{\sqrt{\wh{g}}}\,\pl_i\bigl(\sqrt{\wh{g}}\,X^i\bigr)\Bigr)
=\frac1{\sqrt{\ka_*\wh{g}}}\,\pl_i\bigl(\sqrt{\ka_*\wh{g}}\,Y^i\bigr).
$$
Since
\hb{\sqrt{\ka_*\wh{g}}=\sqrt{\wt{g}}\,\sqrt{\wt{h}}
   =\rho^{-2}\sqrt{\wt{h}}}, we see that
$$
\ka_*(\tdiv_{\wh{g}}X)
=\rho^2\bigl(\pl_x(\rho^{-2}Y^1)+\pl_y(\rho^{-2}Y^2)\bigr)
+\tdiv_{\wt{h}}\wt{Y}.
$$
From this and \eqref{MB.gr} we obtain, letting
\hb{\wt{a}:=\ka_*a},
\begin{equation}\label{MB.kA}
\bal
\ka_*(\cA u)
&=-\ka_*\bigl(\tdiv_{\wh{g}}(a\grad_{\wh{g}}u)\bigr)\cr
&=-\rho^2\bigl(\pl_x(\wt{a}\pl_xv)+\pl_y(\wt{a}\pl_yv)\bigr)
 -\tdiv_{\wt{h}}(\wt{a}\grad_{\wt{h}}v).
\eal
\end{equation}
As in \eqref{R.nu}, we define curvilinear derivatives by
$$
\pl_\nu u:=\ka^*\pl_x(\ka_*u)
\qb \pl_\mu u:=\ka^*\pl_y(\ka_*u).
$$
Then it follows from \eqref{MB.kA} that
\begin{equation}\label{MB.A}
\cA_Uu=-\rho^2\bigl(\pl_\nu(a\pl_\nu u)+\pl_\mu(a\pl_\mu u)\bigr)
-\tdiv_\Sa(a\grad_\Sa u),
\end{equation}
where $\cA_U$~is the restriction of~$\cA$ to~$U$. Due
to~\eqref{F.1}, the regularity assumption for~$a$ stipulated in
Theorem~\ref{thm-R.RD} means that
\hb{a\in\bar{BC}^{1/\mf2}(\wh{M}\ssm S\times J,\ \wh{g}+dt^2)}.

\par
Analogously,
$$
\ka_*(\cB^1u)
=\ka_*\bigl(\nu\bsn\ga(a\grad_{\wh{g}}u)\bigr)_{\wh{g}}
=\Bigl(\ka_*\nu\Bsn
 \ga\bigl(\wt{a}(\grad_{\wt{g}}v
 +\grad_{\wt{h}}v)\bigr)\Bigr)_{\wt{g}+\wt{h}}.
$$
From \eqref{C.nu} and
\hb{(\ka_*\wh{g})^{1i}=0} for
\hb{i\geq2} we deduce that
\hb{\ka_*\nu(0,y,z)=\rho(0,y)e_1}.
Hence
$$
\ka_*(\cB^1u)(y,z)
=\rho(0,y)\wt{a}(0,y,z)\pl_xv(0,y,z).
$$
This implies
\begin{equation}\label{MB.B}
\cB_U^1u=\ga(\rho a)\pl_\nu u
\text{ on }\Ga\cap U.
\end{equation}

\par
Next we determine the first order transmission operator on~$\whMwhg$.
Recalling definition~\eqref{H.f}, we set
\hb{s:=\chi^*f\in C^\iy(U)}.

\setcounter{proposition}{2}
\begin{proposition}\label{pro-MB.S}
Define
\begin{equation}\label{MB.nu}
(\nu_S^1,\nu_S^2,\nu_S^3)
:=(\pl_\nu s,-1,1)
\Big/\sqrt{1+(\pl_\nu s)^2+|\grad_\Sa s|_\Sa^2}.
\end{equation}
Then
$$
\cC_U^1u
=\eea{a\pl_{\nu_S}u}
=\beea{a\bigl(\nu_S^1\pl_\nu u+\nu_S^2\pl_\mu u
 +\nu_S^3(\grad_\Sa u\sn\grad_\Sa s)_\Sa\bigr)}.
$$
\end{proposition}

\begin{proof}
(1) It follows from \eqref{H.f} that
$$
\wt{S}:=\chi(S)
=\bigl\{\,(x,f_\sa(x),\sa)\ ;\ 0<x<\ve,\ \sa\in\Sa\,\bigr\}.
$$
We write
\hb{f(x,z):=f_{\lda^{-1}(z)}(x)}. Then
\hb{(x,z)\mt F(x,z):=\bigl(x,f(x,z),z\bigr)} is a local
parametrization of~$\wt{S}$, and
$$
\pl F:=[\pl_jF^i]
=\left[
\begin{array}{ccc}
1       &\  &0\\
\pl_xf  &\  &\pl_zf\\
0       &\  &1_{m-2}
\end{array}
\right]
\in\BR^{m\times(m-1)}.
$$
Hence, given
\hb{(\bar x,\bar z)\in(0,\ve)\times(-1,1)^{m-2}},
$$
T_{(\bar x,\bar z)}\wt{S}
=\bigl\{(\bar x,\bar z)\bigr\}\times\pl F(\bar x,\bar z)\BR^{m-1}
\is T_{(\bar x,\bar z)}\BR^m.
$$
For
\hb{\Xi:=(\xi,\za)\in\BR\times\BR^{m-2}} and
\hb{\wh{\Xi}:=(\wh{\xi},\wh{\eta},\wh{\za})\in\BR\times\BR\times\BR^{m-2}}
we find
$$
\bigl(\wh{\Xi}\bsn\pl F(\bar x,\bar z)\Xi\bigr)_{\wt{g}+\wt{h}}
=\al(\bar x,\bar z)
\Bigl(\wh{\xi}\xi+\wh{\eta}\bigl(\pl_xf(\bar x,\bar z)\xi
+\pl_zf(\bar x,\bar z)\za\bigr)\Bigr)_{\BR^2}+(\wh{\za}\sn\za)_{\wt{h}},
$$
where
\hb{\al(\bar x,\bar z):=\rho^{-2}\bigl(\bar x,f(\bar x,\bar z)\bigr)}.
Choose
\hb{\wh{\xi}:=\pl_xf(\bar x,\bar z)},
\ \hb{\wh{\eta}:=-1}, and
$\wh{\za}$ in $\BR^{m-2}$ such that
\hb{(\wh{\za}\sn\wt{\za})_{\wt{h}}
   =\bigl\dl\pl_zf(\bar x,\bar z),\wt{\za}\bigr\dr_{\BR^{m-2}}} for all
\hb{\wt{\za}\in\BR^{m-2}}, that is,
$\wh{\za}$~equals $\grad_{\wt{h}}f(\bar x,\bar z)$. Then
\hb{\bigl((\bar x,\bar z),\wh{\Xi}\bigr)\perp T_{(\bar x,\bar z)}\wt{S}}.
Now we define
$$
(\wt{\nu}^1,\wt{\nu}^2,\wt{\nu}^3)
:=(\pl_xf,-1,1)\Big/
\sqrt{1+(\pl_xf)^2+|\grad_{\wt{h}}f|_{\wt{h}}^2}.
$$
Then
$$
\wt{\nu}:=\wt{\nu}^1e_1+\wt{\nu}^2e_2+\wt{\nu}^3\grad_{\wt{h}}f
$$
is the positive\footnote{By the conventions employed in~\eqref{S.nu}.}
normal on~$\wt{S}$, since
\hb{\wt{\nu}^1(\bar x,\bar z)e_1+\wt{\nu}^2(\bar x,\bar z)e_2} is a
positive multiple of the positive normal of the graph of~$f_{\bar\sa}$ at
\hb{\bigl(\bar x,f_{\bar\sa}(\bar x)\bigr)}, where
\hb{\bar\sa:=\lda^{-1}(\bar z)} (cf.~\eqref{S.nu}).

\par
(2)
Using \eqref{MB.gr} and
\hb{\ka_*\nu_S=\wt{\nu}}, we obtain
$$
\bal
\ka_*(\nu_S\sn\grad_{\wh{g}}u)_{\wh{g}}
&=(\ka_*\nu_S\sn\grad_{\wt{g}}v+\grad_{\wt{h}}v)_{\wt{g}+\wt{h}}\cr
&=\wt{\nu}^1\pl_xv+\wt{\nu}^2\pl_yv
 +\wt{\nu}^3(\grad_{\wt{h}}f\sn\grad_{\wt{h}}v)_{\wt{h}}\cr
&=\ka_*\bigl(\nu_S^1\pl_\nu u+\nu_S^2\pl_\mu u
 +\nu_S^3(\grad_\Sa s\sn\grad_\Sa u)_\Sa\bigr),
\eal
$$
the $\nu_S^i$~being given by \eqref{MB.nu}. From this the assertion is now
clear.
\end{proof}

\par
Note that, see~\eqref{MB.W},
\begin{equation}\label{MB.Ga}
\chi_*(d\vol_{\Ga\cap U})\sim\frac{dy}{y^2}\,d\vol_\Sa.
\end{equation}
Since, by~\eqref{H.f},
$$
\wt{g}_{G(\sa)}(\tau)
=\frac{1+(\pl f_\sa(x))^2}{x^2+(f_\sa(x))^2}\,dx^2\sim\frac{dx^2}{x^2}
$$
uniformly with respect to
\hb{\tau=\bigl(x,f_\sa(x)\bigr)\in G_\sa} and
\hb{\sa\in\Sa}, we see that
\begin{equation}\label{MB.S}
\chi_*(d\vol_{S\cap U})\sim\frac{dx}{x^2}\,d\vol_\Sa.
\end{equation}

\par
On the basis of \eqref{MB.Ga} and \eqref{MB.S} it is possible to represent
the trace spaces on
\hb{\Ga\ssm\Sa} and
\hb{S\ssm\Sa} analogously to~$\bar\cW_{\coW p}^{2/\mf2}$.
Details are left to the reader.

\begin{proof}
{\it of Theorem~\ref{R.RD}}
The claim is an easy consequence of Theorems \ref{thm-H.C} and
\ref{thm-L.MR}, \eqref{MB.D1}, \eqref{MB.D2}, Proposition~\ref{pro-MB.N},
\eqref{MB.A}, \eqref{MB.B}, and Proposition~\ref{pro-MB.S}.
\end{proof}

{\footnotesize

config: arxiv