\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2006(2006), No. 118, pp. 1--20.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2006 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2006/118\hfil Dirichlet forms for general Wentzell b.c.]
{Dirichlet forms for general Wentzell boundary conditions,
analytic semigroups, and cosine operator functions}

\author[D. Mugnolo, S. Romanelli \hfil EJDE-2006/118\hfilneg]
{Delio Mugnolo, Silvia Romanelli}  % in alphabetical order

\address{ Delio Mugnolo \newline
Dipartimento di Matematica\\
Universit\`a degli Studi di Bari\\
Via Orabona 4, I-70125 Bari, Italy}
\email{mugnolo@dm.uniba.it}

\address{Silvia Romanelli \newline
Dipartimento di Matematica\\
Universit\`a degli Studi di Bari\\
Via Orabona 4, I-70125 Bari, Italy}
\email{romans@dm.uniba.it}

\dedicatory{Dedicated to Angelo Favini with great admiration and friendship
on his 60-th birthday}

\date{}
\thanks{Submitted June 8, 2006. Published September 28, 2006.}
\subjclass[2000]{47D06, 35J20, 35J25}
\keywords{General Wentzell boundary conditions; Dirichlet forms;
\hfill\break\indent  Ultracontractive semigroups; Cosine operator functions}

\begin{abstract}
 The aim of this paper is to study uniformly elliptic operators
 with general Wentzell boundary conditions in suitable $L^p$-spaces
 by using the approach of sesquilinear forms.
 We use different tools to re-prove analiticity and related results
 concerning the semigroups generated by the above operators.
 In addition, we make some complementary observations on, among
 other things, compactness issues and characterization of domains.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{assum}[theorem]{Assumption}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}

\section{Introduction}

 Favini, G.R. Goldstein, J.A. Goldstein,
 Romanelli \cite{FGGR02} investigated the heat equation in an open
bounded domain $\Omega$ of ${\mathbb R}^n$ governed by the elliptic
operator $A:=\nabla\!\cdot\!(a\nabla)$ with general Wentzell boundary
condition
\begin{equation}\label{GWBC}
Au+\beta\frac{\partial u}{\partial\nu} +\gamma u=0\quad\text{on }
\partial\Omega,
\end{equation}
where $\nu$ is the outer unit normal, and the boundary $\partial\Omega$
is $C^2$. Under the assumption that $a=(a_{ij})=(\alpha\delta_{ij})$
(here $\delta_{ij}$ stands for the identity matrix in ${\mathbb R}^n$)
is a diagonal $n\times n$ matrix, with real valued coefficient
$0<\alpha\in C^1(\overline{\Omega})$, $\beta,\gamma\in C^1(\partial\Omega)$,
 with $\beta>0$, $\gamma\ge 0$ they have considered the realizations
${\mathcal A}_p$ of $A$ in the spaces
$\mathcal{X}^p:=L^p(\overline{\Omega}, d\mu)$, $1\leq p<\infty$,
where $d\mu:=dx|_{_{\Omega}}\oplus\frac{\alpha
d\sigma}{\beta}|_{_{\partial\Omega}}$. The boundary condition can
be rewritten as
$$
\nabla \cdot (a \nabla u) + \frac{\beta}{\alpha}
\big(\alpha\frac{\partial u}{\partial \nu}\big)
+ \gamma u=0\quad\text{on }\partial\Omega,
$$
and $\alpha\frac{\partial u}{\partial \nu}$ is the conormal derivative
 of $u$ with respect to $\alpha$. We refer to~\cite{Go06} for a derivation of such boundary conditions.

Here, $dx$ is the Lebesgue measure on $\Omega$ and $\frac{\alpha
d\sigma}{\beta}$ denotes the natural surface measure $d\sigma$ on
$\partial\Omega$ with weight $\frac{\alpha}{\beta}$. One of the
main results of \cite{FGGR02} was that the operator
$\overline{{\mathcal A}_p}$, $1<p<\infty$, generates an  analytic,
contraction semigroup on $\mathcal{X}^p$, and it is self-adjoint
for $p=2$.

On the other hand,  Favini, G.R. Goldstein, J.A. Goldstein and Romanelli
\cite{FGGR05} also studied a generalization of the above problem to
nonlinear boundary conditions obtaining, in particular, smoothing and
other optimal regularity properties of solutions. Once one specializes
to consider the linear case, these are in fact optimal ultracontractive results.

Arendt, Metafune, Pallara, Romanelli \cite{AMPR03} introduced
the sesquilinear form
\begin{equation}\label{form1}
{\mathcal Q}\left(\begin{pmatrix}u\\ u|_{{\partial\Omega}}\end{pmatrix},
\begin{pmatrix}v\\ v|_{{\partial\Omega}}\end{pmatrix}\right)
:=\int_\Omega (a\nabla u)\!\cdot\!\overline{\nabla v} dx + \int_{\partial\Omega} \gamma u \overline{v} d\sigma
\end{equation}
on $\mathcal{X}^2$ and showed that the operator associated with
such a form is the same $\mathcal A_2$ considered above, provided
that $\alpha=\beta\equiv 1$ and $\partial\Omega$ is Lipschitz.
Thus, they could deduce generation and regularity results (also in
$C(\overline{\Omega})$) by properties of $\mathcal Q$ (see also
\cite{[Wa03]}, in the case of the space $C[0,1]$).

In a similar order of ideas, Vogt and  Voigt \cite{VV03} investigated
 the form $\mathcal Q$ on a slightly different, weighted product Hilbert
space that formally covers the case when the coefficient $a$ may vanish.
They proved that $\mathcal Q$ is Dirichlet even under weaker assumptions
on $a,\beta,\gamma$, and $\partial\Omega$. However, the setting in
\cite{VV03} is very general, so that the identification of the operator
 associated with $ \mathcal Q$ is not easy to perform.

The approach by operator matrices followed by  Engel \cite{En04b} and
by  Xiao and Liang \cite{XL04} has recently revealed that an
elliptic operator in non-divergence form with \eqref{GWBC} generates a
cosine operator function (hence an analytic semigroup) in the space
$C[0,1]$ (see also \cite{BE04}).

Likewise,  Engel and  Fragnelli \cite{EF05} have shown that a uniformly
elliptic second order operator in divergence form with \eqref{GWBC}
generates an analytic semigroup in the space $C(\overline{\Omega})$.

The aim of the present paper is twofold: First we use the theory
of Dirichlet sesquilinear forms as presented, e.g., in
\cite{Da90} and \cite{Ou04}, in order to deduce results
concerning regularity and ultracontractivity of solutions that had
already been obtained in \cite{FGGR05} in a nonlinear context;
second, to present some slight improvements to the known theory.
We consider the operator associated with $\mathcal Q$ and show
that the family of $\mathcal{X}^p$ semigroups, $1<p<\infty$, to
which the semigroup on $\mathcal{X}_2$ extends are analytic with
an angle estimate that improves that obtained in \cite{FGGOR05}.
We then extend to the uniformly elliptic case a compactness result
obtained in \cite{AMPR03},
 which in turn allows to apply the Perron-Frobenius theory.
In addition we obtain some results on uniform convergence to a positive
projection onto the subspace of constant functions or to 0, depending
on the coefficients on the boundary.

Note that if $a$ and $\partial\Omega$ are smooth enough, then the
operator associated with $\mathcal Q$ can be  described rather explicitly, cf.
Theorem \ref{ident}: as it has already been shown in \cite{FGGR05},
such an operator matrix is in fact the realization of the uniformly
elliptic operator $\nabla\!\cdot(a\nabla)$ with general Wentzell boundary
conditions ${\rm \eqref{GWBC}}$.
Of course, all the properties we show for semigroups and cosine operator
functions actually have a counterpart for solutions to suitable first and
second order Cauchy problems.

Much more can be said in the special case of a strictly positive coefficient
$\gamma$ on the boundary. In this framework, in a similar order of ideas as
in the recent monographs \cite{Ar04} and \cite{Ou04}, Section 4 supplies
a treatment of many related results as estimates for ultracontractivity
(particular case of those obtained in \cite{FGGR05} but slightly sharper
than those in \cite{AMPR03}) and spectral properties.


\section{General Framework}

Let $\Omega$ be a bounded open domain of ${\mathbb R}^n$ with
Lipschitz boundary $\partial\Omega$ and $1\leq p\leq \infty$.
Following \cite{FGGR02} (see also \cite{AMPR03}), we introduce
the  product spaces defined by
$$
\mathcal{X}^p:=L^p(\Omega;{\mathbb C})\times
L^p(\partial\Omega;{\mathbb C}), \quad 1\le p\le\infty.
$$
Observe that $\mathcal{X}^p$ can be identified with the space
$L^p(\overline{\Omega}, d\mu)$, equipped with the norm
$$
\Vert\cdot\Vert_{\mathcal{X}^p}:=\left(\Vert\cdot\Vert_{L^p(\Omega)}^p
 + \Vert\cdot\Vert_{L^p(\partial\Omega)}^p\right)^{1/p}\quad
 \hbox{if}\quad 1\leq p<\infty,
$$
or else
$$
\Vert\cdot\Vert_{\mathcal{X}^\infty}:=\Vert\cdot\Vert_{L^\infty(\Omega)}
\vee \Vert\cdot\Vert_{L^\infty(\partial\Omega)}.
$$
(Here $d\mu:=dx|_{\Omega}\oplus d\sigma|_{\partial\Omega}$,  with
$dx$ the Lebesgue measure on $\Omega$ and $d\sigma$ the natural
surface measure on $\partial\Omega$.) Thus, the general theory of
Lebesgue spaces yields
\begin{equation}\label{imbed}
\mathcal{X}^p\hookrightarrow \mathcal{X}^q\quad \hbox{for all }
1\leq q\leq p\leq\infty.
\end{equation}
Moreover, each $\mathcal{X}^p$ is a Banach lattice, and its positive
 cone is the product of the positive cones of $L^p(\Omega)$ and
$L^p(\partial\Omega)$.

Let us consider $\mathcal{X}^2=L^2(\Omega)\times
L^2(\partial\Omega)$. If we equip it with the canonical inner
product
$$
\left<\mathfrak{u},\mathfrak{v}\right>_{\mathcal{X}^2}
:=\left<\begin{pmatrix}u\\w\end{pmatrix},\begin{pmatrix}v\\z\end{pmatrix}\right>_{\mathcal{X}^2}
:=\left<u,v\right>_{L^2(\Omega)}+
\left<w,z\right>_{L^2(\partial\Omega)},
$$
then ${\mathcal{X}^2}$ becomes a Hilbert space. We also define the
linear subspace
$$
\mathcal{V}:=\left\{\begin{pmatrix}u\\ w\end{pmatrix}\in H^1(\Omega)
\times H^{1/2}(\partial\Omega): w=u|_{{\partial\Omega}}\right\}
$$
of $\mathcal{X}^2=L^2(\Omega)\times L^2(\partial\Omega)$. We
emphasize that $\mathcal V$ is \emph{not} a product space.

\begin{lemma}\label{densev}
The linear subspace $\mathcal V$ is dense in $\mathcal{X}^2$.
\end{lemma}

\begin{proof}
It suffices to apply Lemma \ref{lemmadense}, with $X_1=H^1(\Omega)$,
$X_2=L^2(\Omega)$, $Y_1=H^{1/2}(\partial\Omega)$, and
$Y_2=L^2(\partial\Omega)$. Then the boundary trace operator $L$
is bounded from $H^1(\Omega)$ onto $H^{1/2}(\partial\Omega)$,
cf. \cite[Thm.~1.8.3]{LM72}. The claim follows due to the density
of the imbeddings $\ker(L)=H^1_0(\Omega)\hookrightarrow L^2(\Omega)$
 and $H^{1/2}(\partial\Omega)\hookrightarrow L^2(\partial\Omega)$.
\end{proof}

\begin{remark}\label{vhilb} \rm
Observe that
\begin{equation}\label{cannorm}
\left<\mathfrak{u},\mathfrak{v}\right>_{\mathcal{V}}=
\left<\begin{pmatrix}u\\u|_{{\partial\Omega}}\end{pmatrix},
\begin{pmatrix}v\\v|_{{\partial\Omega}}\end{pmatrix}\right>_{\mathcal{V}}:=
\left<u,v\right>_{H^1(\Omega)}+ \left<u|_{{\partial\Omega}},v|_{{\partial\Omega}}\right>_{H^{1/2}(\partial\Omega)},
\end{equation}
defines an inner product on $\mathcal V$. With respect to it,
$\mathcal V$ becomes a Hilbert space.
\end{remark}

\begin{lemma}\label{eqnorm}
The norm $\Vert\cdot\Vert_{\mathcal V}$ on the Hilbert space
$\mathcal V$ defined by \eqref{cannorm} is equivalent to that
defined by
$$
\vert\Vert\mathfrak{u}\Vert\vert^2
=\Big\vert\Big\Vert
\begin{pmatrix}u\\u|_{{\partial\Omega}}\end{pmatrix}\Big\Vert\Big\vert^2
:=\Vert \nabla u\Vert^2_{L^2(\Omega)}+ \Vert u\Vert^2_{L^2(\partial\Omega)},
\quad\mathfrak{u}\in\mathcal{V}.$$
\end{lemma}

\begin{proof}
Due to the boundedness of the boundary trace operator from $H^1(\Omega)$
to $H^{1/2}(\partial\Omega)$, it is enough to apply the inequality
\begin{equation}\label{maz}
\Vert u\Vert_{L^\frac{2n}{n-1}(\Omega)}\leq C(\Vert \nabla
u\Vert_{L^2(\Omega)}+ \Vert u\Vert_{L^2(\partial\Omega)}).
\end{equation}
Such an inequality holds for all $u\in H^1(\Omega)$ by
\cite[Cor.~4.11.2]{Ma85}, cf. also \cite{MV04} for more general
results in this context.
\end{proof}

\section{Main results}

Throughout this paper we impose the following conditions on the coefficients
$a$ and $\gamma$.

\begin{assum}\label{basic} \rm
\begin{enumerate}
\item $a=(a_{ij})$ is a symmetric matrix of real valued
$H^1_{\rm loc}(\Omega)$-functions such that the ellipticity condition
$$
c_1\vert \xi\vert^2\leq {\rm Re}\sum_{i,j=1}^n a_{ij}(x)\xi_i\overline{\xi_j}
\leq C_1\vert\xi\vert^2
$$
holds for suitable constants $0<c_1,C_1$ and all $\xi\in{\mathbb C}^n$,
a.e. $x\in\Omega$.
\item $\gamma\in L^\infty(\partial\Omega)$ is real valued and such
that $0\leq \gamma\leq C_2$ a.e. for a suitable constant $C_2$.
\end{enumerate}
\end{assum}

We define a sesquilinear form $\mathcal Q$ with form domain
$D(\mathcal{Q}):={\mathcal V}$ on the Hilbert space
$\mathcal{X}^2$ by
\begin{equation*}
{\mathcal Q}({\mathfrak u},{\mathfrak v})={\mathcal Q}\left(\begin{pmatrix}u\\
u|_{{\partial\Omega}}\end{pmatrix},\begin{pmatrix}v\\ v|_{{\partial\Omega}}
\end{pmatrix}\right)
:=\int_\Omega (a\nabla u)\!\cdot\!\overline{\nabla v} dx
+ \int_{\partial\Omega} \gamma u \overline{v} d\sigma.
\end{equation*}

Our main aim is to show that $\mathcal Q$ (which is sesquilinear by definition,
 and densely defined by Lemma \ref{densev}) is associated to a submarkovian
semigroup. This is a consequence of the following.

\begin{theorem}\label{main1}
The densely defined sesquilinear form $\mathcal Q$ is symmetric, positive,
and closed, i.e.,
\begin{itemize}
\item ${\mathcal Q}({\mathfrak u},{\mathfrak v})=\overline{{\mathcal Q}({\mathfrak v},{\mathfrak u})}$ for all ${\mathfrak u},{\mathfrak v} \in {\mathcal V}$,

\item ${\mathcal Q}({\mathfrak u},{\mathfrak u})\geq 0$ for all ${\mathfrak u} \in {\mathcal V}$,

\item $\mathcal V$ is complete for the form norm
\begin{align*}
\Vert \mathfrak{u}\Vert_{\mathcal Q}
&:= \sqrt{\mathcal{Q}(\mathfrak{u},\mathfrak{u})+\Vert \mathfrak{u}\Vert^2_{\mathcal{X}^2}}\\
&=\left(\int_\Omega (a \nabla u)\!\cdot\!\overline{\nabla u} dx
 + \int_\Omega \vert u\vert^2 dx + \int_{\partial\Omega} (1+\gamma)
\vert u\vert^2 d\sigma\right)^{1/2}.
\end{align*}
\end{itemize}
\end{theorem}

\begin{proof}
The fact that the coefficients $a$ and $\gamma$ are real-valued yields
that $\mathcal Q$ is symmetric.

To show the positivity of $\mathcal Q$, take
${\mathfrak u}=\begin{pmatrix}u\\u|_{{\partial\Omega}}\end{pmatrix}
\in{\mathcal V}$ and observe that
$$
\mathcal{Q}({\mathfrak u},{\mathfrak u})=\int_\Omega (a\nabla u)\cdot
\overline{\nabla u} dx + \int_{\partial\Omega} \gamma \vert u\vert^2
d\sigma\geq 0,
$$
due to the positivity of $a$ and $\gamma$.

Further, for all ${\mathfrak u}= \begin{pmatrix}u\\u|_{{\partial\Omega}}
\end{pmatrix}\in{\mathcal V}$ one has by assumption
$$
\Vert{\mathfrak u}\Vert^2_{\mathcal Q}\geq c_1\Vert \nabla u\Vert^2_{L^2(\Omega)} +
\Vert u\Vert^2_{L^2(\Omega)} + \Vert u\Vert^2_{L^2(\partial\Omega)}
$$
and
$$
\Vert{\mathfrak u}\Vert^2_{\mathcal Q}\leq C_1\Vert \nabla u\Vert^2_{L^2(\Omega)} +
\Vert u\Vert^2_{L^2(\Omega)} + (1+C_2)\Vert u\Vert^2_{L^2(\partial\Omega)}.
$$
Taking into account Lemma \ref{eqnorm}, we conclude that the form norm of $\mathcal Q$
is equivalent to $\Vert\cdot \Vert_{\mathcal V}$, with respect to which
 the space $\mathcal V$ is complete. Thus, $ \mathcal Q$ is closed.
\end{proof}

\begin{remark}\label{cont} \rm
(1) Observe that the form $\mathcal Q$ is, in fact, also continuous, i.e.,
$$
\vert {\mathcal Q}({\mathfrak u},{\mathfrak v})\vert \leq
M\Vert {\mathfrak u}\Vert_{\mathcal V} \Vert {\mathfrak v}\Vert_{\mathcal V}
$$
for all ${\mathfrak u},{\mathfrak v}\in D({\mathcal Q})$, where
$M:=C_1 \vee C_2<\infty$.

To see this, take into account Remark \ref{vhilb} and Lemma \ref{eqnorm}. By
 the Cauchy--Schwartz inequality and the hypotheses on $a$ one has
$$
\big\vert\sum_{i,j=1}^n a_{ij}(x)\xi_i \overline{\eta_j}\big\vert
\leq \Big(\sum_{i,j=1}^n a_{ij}(x)\xi_i \overline{\xi_j}\Big)^{1/2}
\Big(\sum_{i,j=1}^n a_{ij}(x)\eta_i \overline{\eta_j}\Big)^{1/2}
$$
for all $\xi,\eta\in{\mathbb C}^n$ and a.e. $x\in\Omega$. It
follows that
\begin{align*}
\vert{\mathcal Q}({\mathfrak u},{\mathfrak v})\vert
&= \left\vert\int_\Omega (a\nabla u)\!\cdot\!\overline{\nabla v} dx + \int_{\partial\Omega} \gamma u \overline{v} d\sigma\right\vert\\
&\leq \int_\Omega \vert(a\nabla u)\!\cdot\!\overline{\nabla v}\vert dx + \int_{\partial\Omega} \vert \gamma u\overline{v}\vert d\sigma\\
&\leq  \int_\Omega \left((a\nabla u)\!\cdot\!\overline{\nabla u}\right)^{1/2} \left((a\nabla v)\!\cdot\!\overline{\nabla v}\right)^{1/2} dx + \int_{\partial\Omega} \vert \gamma u\overline{v}\vert d\sigma\\
&\leq  \int_\Omega C_1^{1/2}\vert \nabla u\vert C_1^{1/2} \vert\nabla v\vert dx +  \int_{\partial\Omega} C_2\vert u\vert \vert{v}\vert d\sigma\\
& = C_1 \left<\vert\nabla u\vert,\vert\nabla v\vert\right>_{L^2(\Omega)} + C_2 \left<\vert u\vert,\vert v\vert\right>_{L^2(\partial\Omega)}\\
&\leq  C_1 \Vert \nabla u\Vert_{L^2(\Omega)} \Vert \nabla v\Vert_{L^2(\Omega)} + C_2 \Vert  u\Vert_{L^2(\partial\Omega)}\Vert v\Vert_{L^2(\partial\Omega)}\\
&\leq  M\left( \Vert  u\Vert_{H^1(\Omega)} + \Vert u\Vert_{L^2(\partial\Omega)}\right)
\left( \Vert  v\Vert_{H^1(\Omega)} + \Vert v\Vert_{L^2(\partial\Omega)}\right)\\
&=M\Vert{\mathfrak u}\Vert_{\mathcal V} \Vert {\mathfrak v}\Vert_{\mathcal V}.
\end{align*}
(2) The form $\mathcal Q$ is also local, i.e.,
$\mathcal{Q}(\mathfrak{u},\mathfrak{v})=0$ for all
$\mathfrak{u},\mathfrak{v}\in \mathcal{V}$ such that
$\mathop{\rm supp}(\mathfrak{u})\cap \mathop{\rm supp}(\mathfrak{v})$
is a $\mu$-null set.
\end{remark}

Thus, one can associate to $\mathcal Q$ an operator
$\mathcal{A}_2$ on $\mathcal{X}^2$, given by
\begin{align*}
D({\mathcal A}_2)&:=\left\{{\mathfrak u}\in {\mathcal V}:
\exists {\mathfrak z}\in {\mathcal{X}^2} \hbox{ s.t. }
{\mathcal Q}({\mathfrak u},{\mathfrak v})
=\left<{\mathfrak z},{\mathfrak v}\right>_{\mathcal{X}^2}\;
 \forall {\mathfrak v}\in {\mathcal V}\right\},\\
{\mathcal A}_2{\mathfrak u}&:=-{\mathfrak z}.
\end{align*}
By \cite[Thm.~1.2.1]{Da90}, such an operator is self-adjoint and dissipative.
In fact, the following holds.

\begin{proposition}\label{cosine}
The operator ${\mathcal A}_2$ associated with $\mathcal Q$
generates a cosine operator function with associated phase space
$\mathcal{V}\times\mathcal{X}^2$.
\end{proposition}

\begin{proof}
Due to the bounded perturbation theorem for cosine operator
functions (see Lemma \ref{pert}), the claim follows if we show
that ${\mathcal A}_2+I_{\mathcal{X}^2}$ generates a cosine
operator function with the same associated phase space. Define now
the form
$$
\tilde{\mathcal Q}(\mathfrak{u},\mathfrak{v}):=\mathcal{Q}(\mathfrak{u},\mathfrak{v})+
\left<\mathfrak{u},\mathfrak{v}\right>_{\mathcal{X}^2},\quad
\mathfrak{u},\mathfrak{v}\in \mathcal V.
$$
on $\mathcal{X}^2$. It is apparent that $\tilde{\mathcal Q}$
is sesquilinear, densely defined, symmetric, and closed.
Moreover, it is bounded below by
1, and the associated operator is exactly ${\mathcal
A}_2+I_{\mathcal{X}^2}$. Thus, the claim follows by
\cite[Prop.~7.1.3]{ABHN01}.
\end{proof}

\begin{remark}\rm
 Favini, G.R. Goldstein, J.A. Goldstein,  Romanelli \cite[Theorem 3.1]{FGGR02}
showed that if $\Omega$ has boundary $C^2$, $a\in C^1(\overline{\Omega})$,
$a>0$ in $\Omega$, and $\Gamma=\{z\in\partial\Omega: \, a(z)>0\}\neq\emptyset$,
 then the closure of the realization on
$L^2(\Omega, dx)\oplus L^2 (\Gamma, \frac{a\, d\sigma}{\beta})$
of the operator $A:=\nabla\cdot (a\nabla)$ with general Wentzell boundary
condition
$$
A u+\beta\frac{\partial u}{\partial\nu} +\gamma u=0\quad \hbox{on} \quad \Gamma
$$
is self-adjoint and dissipative, hence it generates a cosine operator function.
There $\beta, \gamma$ were assumed to be in $C^1(\partial\Omega)$, with
$\beta>0$ and $\gamma\ge 0$ on $\partial\Omega$.
Recently,  Favini, G.R. Goldstein, J.A. Goldstein,  Obrecht, and Romanelli
\cite{FGGOR05} have extended this result to the case of a more general
elliptic operator of the type
$$
A:=\sum_{i,j=1}^n \frac{\partial}{\partial x_i}
\left(a_{ij}\frac{\partial}{\partial x_j}\right)
$$
with general Wentzell boundary condition given by
$$
Au+\beta\frac{\partial_a u}{\partial\nu} +\gamma u=0\hbox{ on }\Gamma.
$$
Here $\frac{\partial_a u}{\partial\nu}$ denotes the conormal derivative
 of $u$ with respect to $a=(a_{ij})$.
\end{remark}

Since $\mathcal{A}_2$ is self-adjoint and dissipative, it also generates
 a strongly continuous semigrop $\mathcal{T}_2$ that is contractive
and analytic of angle~$\frac{\pi}{2}$. In fact, much more can be said
about $\mathcal{T}_2$.

\begin{theorem}\label{main}
The semigroup $\mathcal{T}_2$ on $\mathcal{X}^2$ associated with
$\mathcal Q$ is sub-Markovian, i.e., it is real, positive, and
contractive on $\mathcal{X}^\infty$.
\end{theorem}

\begin{proof}
By \cite[Prop.~2.5, Thm.~2.7, and Cor.~2.17]{Ou04}, we need to check
that the following criteria are verified:
\begin{itemize}
\item ${\mathfrak u}\in \mathcal{V} \Rightarrow \overline{\mathfrak u}\in \mathcal{V} \hbox{ and } {\mathcal Q}({\rm Re}\, {\mathfrak u},{\rm Im}\, {\mathfrak u})\in\mathbb{R}$,
\item ${\mathfrak u}\in \mathcal{V},\; \mathfrak{u}\hbox{ real-valued }\Rightarrow \vert {\mathfrak u}\vert \in \mathcal{V} \hbox{ and } {\mathcal Q}(\vert{\mathfrak u}\vert,\vert{\mathfrak u}\vert)\leq {\mathcal Q}(\mathfrak{u},\mathfrak{u})$,
\item $0\leq {\mathfrak u}\in {\mathcal V} \Rightarrow 1\wedge {\mathfrak u}\in \mathcal{V}  \hbox{ and } {\mathcal Q}(1\wedge {\mathfrak u},(\mathfrak{u}-1)^+)\geq 0$.
\end{itemize}

It is clear that $\overline{u}\in H^1(\Omega)$ if $u\in H^1(\Omega)$,
 hence if ${\mathfrak u}=\begin{pmatrix}u\\ u|_{{\partial\Omega}}\end{pmatrix}
\in \mathcal{V}$, then also
 $\overline{\mathfrak u}=\begin{pmatrix}\overline{u}\\
\overline{u}|_{{\partial\Omega}}\end{pmatrix}\in \mathcal{V}$.
Moreover, ${\mathcal Q}({\rm Re}\,{\mathfrak u},{\rm Im}\,{\mathfrak u})$
is the sum of two integrals. Since all the integrated functions are
real-valued, it follows that
${\mathcal Q}({\rm Re}\, {\mathfrak u},{\rm Im}\, {\mathfrak u})\in\mathbb R$.

To check the second condition let $u\in H^1(\Omega)$.
Then $|u||_{{\partial\Omega}}=|u|_{{\partial\Omega}}|$.
Moreover $\nabla |u|=({\rm sign}\, u)\nabla u$ (see \cite[\S~7.6]{GT77}).
Hence
$$
Q(|u|,|u|)=\int_{\Omega} (a\nabla u)\!\cdot\!\overline{\nabla u} \, dx
+\int_{\partial\Omega} \gamma |u|^2\, d\sigma=Q(u,u).
$$

Finally, as in the point (d) in the proof of \cite[Thm.~2.3]{AMPR03} we
see that if $0\leq u\in H^1(\Omega)$, then $1\wedge u\in H^1(\Omega)$ and
$\nabla\!\cdot\!(1\wedge u)={1}_{\{u<1\}}\nabla u$, while
$\nabla\left((u- 1)^+\right)={1}_{\{u>1\}}\nabla u$, i.e.,
$\nabla\!\cdot\!(1\wedge u)$ and $\nabla\left((u- 1)^+\right)$ are disjointly
supported. Further, if
$\mathfrak{u}=\begin{pmatrix}u\\ u|_{{\partial\Omega}}\end{pmatrix}
\in \mathcal{V}$, then
$$
1\wedge\mathfrak{u}=\begin{pmatrix}1\wedge u\\
(1\wedge u)|_{{\partial\Omega}}\end{pmatrix}\in \mathcal{V}
\quad\hbox{and}\quad (\mathfrak{u}- 1)^+
=\begin{pmatrix}(u- 1)^+\\ ((u- 1)^+)|_{{\partial\Omega}}\end{pmatrix}.
$$
It follows that if $(x,z)\in{\rm supp}(\mathfrak{u}\wedge 1)\cap
{\rm supp}\left((\mathfrak{u}- 1)^+\right)$, then necessarily $u(z)=1$,
and the claim follows by positivity of the coefficient $\gamma$.
\end{proof}

\begin{remark}\label{domin} \rm
Let $\gamma,\tilde{\gamma}$ be functions on $\Omega$ and $\partial\Omega$
satisfying the Assumptions \ref{basic}. Denote by $\mathcal{Q}_{\gamma}$,
$\mathcal{Q}_{\tilde{\gamma}}$ the form $\mathcal Q$ with coefficients
$\gamma$ and $\tilde{\gamma}$, respectively, and by $\mathcal{T}_{\gamma}$,
$\mathcal{T}_{\tilde{\gamma}}$ the associated sub-Markovian
$\mathcal{T}_{2}$-semigroups.
Then $D(\mathcal{Q}_{\gamma})=D(\mathcal{Q}_{\tilde{\gamma}})=\mathcal V$. Also, $\mathcal V$ is an ideal of itself by \cite[Prop.~2.20]{Ou04}.
Assume now that $\gamma(z)\leq \tilde{\gamma}(z)$ for a.e. $z\in\partial\Omega$. A direct computation shows that
$$
\mathcal{Q}_{\gamma}(\mathfrak{u},\mathfrak{v})
\leq \mathcal{Q}_{\tilde{\gamma}}(\mathfrak{u},\mathfrak{v})
$$
for all $0\leq \mathfrak{u},\mathfrak{v} \in{\mathcal V}$, and it
then follows from \cite[Thm.~2.24]{Ou04} that $\mathcal{T}_{\tilde{\gamma}}$
is dominated by $\mathcal{T}_{{\gamma}}$ in the sense of positive semigroups,
 i.e.,
$$
\vert \mathcal{T}_{\tilde{\gamma}}(t)\mathfrak{f}\vert
\leq \mathcal{T}_{{\gamma}}(t)\vert \mathfrak{f}\vert
\quad\hbox{for all\;}\mathfrak{f}\in \mathcal{X}^2,\; t\geq 0.
$$
\end{remark}

The more general case of non-positive $\gamma$ will be treated later on
in this section (see Corollaries \ref{gamma1} and \ref{gamma2}).

\begin{lemma}\label{ultralemma}
The semigroup ${\mathcal T}_2$ on $\mathcal{X}^2$ associated with
$\mathcal Q$ is ultracontractive, i.e., it satisfies the estimate
\begin{equation}\label{ultra}
\Vert{\mathcal T}_2(t){\mathfrak f}\Vert_{\mathcal{X}^\infty} \leq
M_\mu t^{-\frac{\mu}{4}}\Vert{\mathfrak f}\Vert_{\mathcal{X}^2}
\quad\hbox{ for all }t\in (0,1],\; {\mathfrak
f}\in{\mathcal{X}^2},
\end{equation}
for
\begin{equation*}\label{range}
\mu\in
\begin{cases}
[2n-2,\infty), &\hbox{ if }\; n\geq 3,\\
(2,\infty), &\hbox{ if }\; n=2,\\
[1,\infty), &\hbox{ if }\; n=1,\\
\end{cases}
\end{equation*}
and some constant $M_\mu$.
\end{lemma}

\begin{proof}
By \cite[Cor.~2.4.3]{Da90} it suffices to show
that
\begin{equation}\label{ultra1}
\Vert \mathfrak{u}\Vert^2_{\mathcal{X}^\frac{2\mu}{\mu-2}}
\leq M_\mu\Vert\mathfrak{u}\Vert_{\mathcal Q}^2
\end{equation}
for some $\mu>2$ and some constant $M_\mu$.

Take $n\geq 3$ and recall that by the usual Sobolev imbedding theorem we obtain
\begin{equation}\label{sobolev}
\begin{aligned}
\Vert u\Vert_{L^\frac{2\mu}{\mu-2}(\Omega)}
&\leq M_1\Vert u\Vert_{L^\frac{2n}{n-2}(\Omega)}\\
&\leq M_2 \left(\Vert \nabla u\Vert^2_{L^2(\Omega)} + \Vert
u\Vert^2_{L^2(\Omega)}\right) ^{\frac{1}{2}}, \quad u\in
H^{1}(\Omega),
\end{aligned}
\end{equation}
where we have set $\mu=2n-2$, cf. \cite[(7.30)]{GT77}. On the other hand,
by \cite[Theorem~2.4.2]{Ne67} there holds
$$
\Vert u\Vert_{L^\frac{2n-2}{n-2}(\partial\Omega)}
\leq M_3 \left(\Vert \nabla u\Vert^2_{L^2(\Omega)}
+ \Vert u\Vert^2_{L^2(\Omega)}\right)^{\frac{1}{2}},
\quad u\in H^{1}(\Omega);
$$
or rather,
\begin{equation}\label{necas}
\Vert u\Vert_{L^\frac{2\mu}{\mu-2}(\partial\Omega)}\leq M_3
\left(\Vert \nabla u\Vert^2_{L^2(\Omega)} + \Vert
u\Vert^2_{L^2(\Omega)}\right)^{\frac{1}{2}}, \quad u\in
H^{1}(\Omega).
\end{equation}
Combining \eqref{sobolev} and \eqref{necas} yields the claimed
inequality for $\mu=2n-2$, due to the Assumption \ref{basic}.(1).
Taking into account \eqref{imbed} yields \eqref{ultra1} for $\mu>2n-2$.

If $n\leq 2$, then again by \cite[(7.30)]{GT77} and by
\cite[Theorem 2.4.6]{Ne67} the inequalities  \eqref{sobolev} and
 \eqref{necas} prevail for arbitrary $\mu$, \eqref{ultra1} holds
again for $\mu>2$ and the claim follows.

Consider finally the case $n=1$, $\mu\in [1,2]$. In this case it is more convenient to use a criterion based on a Nash-type inequality. In fact, by \cite[Cor.~2.4.7]{Da90} it suffices to show that for all $0\leq \mathfrak{u}\in\mathcal{V}$ there holds
\begin{equation}\label{ultra1b}
\Vert \mathfrak{u}\Vert_{\mathcal{X}^2}\leq
M_\mu \Vert\mathfrak{u}\Vert_{\mathcal Q}^{\frac{\mu}{\mu+2}} \cdot \Vert\mathfrak{u}\Vert^{\frac{2}{\mu+2}}_{\mathcal{X}^1},
\end{equation}
for all $\mu\geq 1$ and some constant $M_\mu$. Recall the inequality
\begin{equation}\label{nash}
\Vert u\Vert_{L^{\frac{2}{2-3\tau}}(0,1)}
\leq M_4 \left(\Vert u'\Vert_{L^2(0,1)}+\Vert u\Vert_{L^1(0,1)}\right)^\tau
\cdot \Vert u\Vert_{L^1(0,1)}^{1-\tau},
\end{equation}
which is valid for all $\tau\in [0,\frac{2}{3}]$ and some constant $M_4$,
cf. \cite[Thm.~1.4.8.1]{Ma85}. Take now $\mu\in [1,2]$ and set
$\tau:=\frac{\mu}{\mu+2}$, so that
$\frac{2}{2-3\tau}=\frac{2\mu +4}{4-\mu}\geq 2$. It follows
by \eqref{nash} that
\begin{equation}\label{nashapp}
\Vert u\Vert_{L^2(0,1)}\leq M_5 \left(\Vert u'\Vert_{L^2(0,1)}
+\Vert u\Vert_{L^2(0,1)}\right)^\frac{\mu}{\mu+2} \cdot
 \Vert u\Vert_{L^1(0,1)}^\frac{2}{\mu+2}.
\end{equation}
Finally, observe that in the case $n=1$ we have
$L^p(\partial\Omega)=\mathbb{C}^2$, $1\leq p\leq \infty$,
so that all the norms on $L^p(\partial\Omega)$ are equivalent.
This and \eqref{nashapp} yield \eqref{ultra1b}.
\end{proof}

\begin{remark} \rm
Following Varopoulos (\cite[\S~0.1]{Va85}, cf. also
\cite[\S~7.3.2]{Ar04}) the number
$$
\mathop{\rm dim}(\mathcal{T}_2):=\inf \{\mu > 0 : \eqref{ultra1}
\hbox{ is valid for some } M_\mu\}
$$
is sometimes called the dimension of the semigroup $\mathcal{T}_2$.
 Hence, we have shown that
\begin{equation*}
\mathop{\rm dim}(\mathcal{T}_2)\leq
\begin{cases}
2n-2, &\hbox{if }\; n\geq 2,\\
1, &\hbox{if }\; n=1.\\
\end{cases}
\end{equation*}
Observe that this improves an analogous result in \cite{AMPR03},
 where in the case $n=1$ it has only been shown that
$\mathop{\rm dim}(\mathcal{T}_2)\leq 2$. Moreover, under slightly
stronger assumptions on the coefficients $a,\gamma$, it was shown
in \cite{FGGR05} that the dimension of ${\mathcal T}_2$ is always $n$.
\end{remark}

Due to the boundedness of $\Omega$, the following holds by
 \cite[Thm.~1.4.1, Thm.~2.1.4, and Thm.~2.1.5]{Da90} and
\cite[Thm.~3.13]{Ou04}.

\begin{corollary}\label{comp}
The semigroup $\mathcal{T}_2$ extends to a family of compact,
contractive, real, positive one-parameter semigroups
${\mathcal T}_p$ on $\mathcal{X}^p$, $1\leq p\leq \infty$. Such semigroups
are strongly continuous if $p\in[1,\infty)$, and analytic of angle
$\frac{\pi}{2}-\arctan\frac{\vert p-2\vert}{2\sqrt{p-1}}$ for
$p\in(1,\infty)$.

Moreover, the spectrum of ${\mathcal A}_p$ is independent of $p$,
where $\mathcal{A}_p$ denotes the generator of ${\mathcal T}_p$.
All the eigenfunctions of $\mathcal{A}_2$ are of class $\mathcal{X}^\infty$.
\end{corollary}

\begin{remark}\label{integral} \rm
(1) As a direct consequence of the ultracontractivity of ${\mathcal T}_2$,
 it follows by \cite[Lemma~2.1.2]{Da90} that ${\mathcal T}_2$ has an
integral kernel ${\mathcal K}$ such that
\begin{equation}\label{kern}
0\leq {\mathcal K}(t,{\bf x},{\bf y})\leq M^2_\mu
t^{-\mu/2}\quad \hbox{for all }t> 0,\;\hbox{ a.e. }
 {\bf x},{\bf y}\in\overline{\Omega},
\end{equation}
where $\mu$ and $M_\mu$ are the same parameters that appear in \eqref{ultra}.

(2) Since the operator $\mathcal A$ is self-adjoint and dissipative,
its spectrum is contained in the negative halfline. Moreover, by the
above corollary the spectral bounds of all operators
${\mathcal A}_p$, $p\in[1,\infty)$, agree. Since the growth bound of a
positive semigroup on an $L^p$-space agrees with the spectral bound of
its generator (see \cite[Thm.~1]{We95}), we conclude that
$s({\mathcal A})$ is the common growth bound of all the semigroups
${\mathcal T}_p$, $p\in[1,\infty)$. Observe that, as an application of
the abstract spectral theory for so-called one-sided coupled operator
matrices developed by K.-J. Engel in \cite{En99}, such a spectral
 bound can in several concrete cases be explicitely computed
(see \cite[\S~9]{KMN03} for a one-dimensional example).
\end{remark}

We still need to identify the operator associated with $\mathcal Q$.

\begin{theorem}\label{ident}
Let $\partial\Omega\in C^\infty$ and $a_{ij}\in C^\infty(\overline{\Omega})$,
$1\leq i,j\leq n$. Then the operator $\mathcal{A}_2$ associated with
the Dirichlet form $\mathcal Q$ is given by
\begin{gather*}
D(\mathcal{A}_2)=\left\{\begin{pmatrix} u\\ w\end{pmatrix}\in H^\frac{3}{2}(\Omega)
\times H^1(\partial\Omega): w=u|_{{\partial\Omega}}\hbox{ and }
\nabla\!\cdot\!(a\nabla u)\in L^2(\Omega) \right\},\\
\mathcal{A}_2=\begin{pmatrix} \nabla\!\cdot\!(a\nabla) & 0\\
-\langle a\nabla,\nu\rangle  & -\gamma I\end{pmatrix}.
\end{gather*}
\end{theorem}

Here $\langle a\nabla u,\nu\rangle$ denotes the conormal derivative of $u$
 with respect to $a$, which is well defined (in the sense of traces)
as an element of $L^2(\partial\Omega)$ for $u\in H^\frac{3}{2}(\Omega)$
due to the regularity of $a$, cf. \cite[\S~2.7]{LM72}.

\begin{proof}
Observe first that we only need to prove the claim for $\gamma\equiv0$, since
$$
\begin{pmatrix} \nabla\!\cdot\!(a\nabla) & 0\\
-\langle a\nabla,\nu\rangle  & -\gamma I\end{pmatrix}=
\begin{pmatrix} \nabla\!\cdot\!(a\nabla) & 0\\
-\langle a\nabla,\nu\rangle  & 0\end{pmatrix}+
\begin{pmatrix} 0 & 0\\
0 & -\gamma I\end{pmatrix},
$$
where the second addend on the
right-hand side is a bounded operator that does not affect the
domain of the first one.

The operator associated with $\mathcal Q$ is, by definition,
\begin{align*}
D({\mathcal B})&:=\left\{{\mathfrak u}\in {\mathcal V}:
\exists {\mathfrak z}\in {\mathcal{X}^2} \hbox{ s.t. }
 {\mathcal Q}({\mathfrak u},{\mathfrak v})=\left<{\mathfrak z},{\mathfrak v}\right>_{\mathcal{X}^2}\; \forall {\mathfrak v}\in {\mathcal V}\right\},\\
{\mathcal B}{\mathfrak u}&:=-{\mathfrak z}.
\end{align*}

To see that $\mathcal{A}_2\subset \mathcal{B}$, observe first that
$D(\mathcal{A}_2)\subset \mathcal{V}$. Take
$\mathfrak{u}=\begin{pmatrix}u\\ u|_{{\partial\Omega}}\end{pmatrix}$ and
$\mathfrak{v}=\begin{pmatrix}v\\ v|_{{\partial\Omega}}\end{pmatrix}$
in $\mathcal{V}$ and apply the Gauss--Green formula to obtain
\begin{align*}
{\mathcal Q}({\mathfrak u},{\mathfrak v})
&=\int_{\Omega} (a\nabla u)\!\cdot\!\overline{\nabla v} dx\\
&= -\int_{\Omega} \nabla\!\cdot\!(a\nabla u) \overline{v} dx
  + \int_{\partial\Omega} \langle a\nabla u,\nu\rangle  \overline{v} d\sigma\\
&= \left< \begin{pmatrix}-\nabla\!\cdot\!(a\nabla u)\\
\langle a\nabla u,\nu \rangle \end{pmatrix},
\begin{pmatrix}v\\ v|_{{\partial\Omega}}\end{pmatrix}\right>_{\mathcal{X}^2}
=\left< -\mathcal{A}_2\mathfrak{u},\mathfrak{v}\right>_{\mathcal{X}^2}.
\end{align*}

Conversely, let $\mathfrak{u}=\begin{pmatrix}u\\
u|_{{\partial\Omega}}\end{pmatrix}\in D(\mathcal{B})$ and repeat the
above computation to obtain that $\nabla\!\cdot\!(a\nabla u)$ is well
defined as an element of $L^2(\Omega)$ and that $u$ has a conormal
derivative (in the sense of traces) in $L^2(\partial\Omega)$.
Thus, by \cite[Thm.~2.7.4]{LM72} we deduce that $u\in H^\frac{3}{2}(\Omega)$.
Since the second entry of the vector ${\mathfrak u}$ is the trace of a
function of class $H^\frac{3}{2}(\Omega)$, it follows by usual boundary
regularity results that $u_{|\partial\Omega}\in H^1(\partial\Omega)$,
and hence $\mathfrak{u}\in D(\mathcal{A}_2)$.
\end{proof}

\begin{remark} \label{rmk3.13} \rm
The strong regularity assumption on $\partial\Omega$ and on the coefficient
 $a$ in Theorem \ref{ident} is solely necessary in order to apply the
results in \cite{LM72} on the regularity of solutions to a Neumann-type
problem.
\end{remark}

We are now in the position to discuss the complete second order operators.

\begin{corollary}\label{gamma1}
Under the assumptions of Theorem \ref{ident}, let
$b\in \left(L^2(\Omega)\right)^n$, $c\in L^2(\Omega)$, and
$\tilde{\gamma}\in L^2(\partial\Omega)$. Then the operator
 $\tilde{\mathcal A}_2$ defined by
\begin{equation*}
\tilde{\mathcal A}_2
:=\begin{pmatrix} \nabla\!\cdot\!(a\nabla) +b\!\cdot\!\nabla +cI & 0\\
-\langle a\nabla,\nu\rangle  & -\tilde{\gamma} I\end{pmatrix}
\end{equation*}
with domain $D(\tilde{\mathcal A}_2):=D({\mathcal A}_2)$ generates
a cosine operator function with associated phase space
$\mathcal{V}\times\mathcal{X}^2$, hence also an analytic semigroup
of angle $\frac{\pi}{2}$ on $\mathcal{X}^2$.
\end{corollary}

\begin{proof}
Write $\tilde{\mathcal A}_2$ as
$$\tilde{\mathcal A}_2={\mathcal A}_2+\mathcal{B}_2,
$$
where
\begin{equation*}
{\mathcal B}_2:=\begin{pmatrix} b\!\cdot\!\nabla +cI & 0\\
0 & (\gamma-\tilde{\gamma}) I\end{pmatrix}.
\end{equation*}
Since by assumption $b\!\cdot\!\nabla +cI$ is bounded from
$H^1(\Omega)$ to $L^2(\Omega)$ and $(\gamma-\tilde{\gamma}) I$ is
bounded on $L^2(\partial\Omega)$ it is clear that ${\mathcal B}_2$
is bounded from $\mathcal V$ to $\mathcal{X}^2$. Now the claim
follows by Lemma \ref{pert}.
\end{proof}

In the introduction we have claimed that the operator matrix associated
with the sesquilinear form $ \mathcal Q$, i.e., $ {\mathcal A}_2$,
is in fact a realization of a second order elliptic operator in divergence
form with general Wentzell boundary conditions. Recalling that
$D({\mathcal A}^2_2)$ is a core for ${\mathcal A}_2$, this is made clear
 in the following.

\begin{corollary}\label{wentz}
Under the assumptions of Theorem \ref{ident}, let  $b\in \left(L^2(\Omega)\right)^n$, $c\in L^2(\Omega)$, and $\tilde{\gamma}\in L^2(\partial\Omega)$. Then for all $u$ such that $\begin{pmatrix}u\\ u|_{{\partial\Omega}}\end{pmatrix}$ is in the domain of $\tilde{\mathcal A}_2^2$ there holds
\begin{equation}\label{went}
\begin{array}{ll}
&\nabla\!\cdot\!\left(a(z)\nabla u(z)\right)+b(z)\!\cdot\!\nabla u(z)+c(z)u(z)\\
&\quad +\langle a(z)\nabla u(z),\nu(z)\rangle +\tilde{\gamma}(z) u(z)=0
\end{array}
\quad\hbox{ for all } z\in\partial\Omega.
\end{equation}
\end{corollary}

This is equivalent to the notion of of weak solution given in \cite{FGGR02}.

\begin{proof}
Take $u$ such that
$\begin{pmatrix}u\\ u|_{{\partial\Omega}}\end{pmatrix}
=:\mathfrak{u}\in D(\tilde{\mathcal A}_2^2)$. Then by definition
$$
\begin{pmatrix} \nabla\!\cdot\!(a\nabla u) +b\!\cdot\!\nabla u +cu\\
-\langle a\nabla u,\nu\rangle  -\tilde{\gamma}
u|_{{\partial\Omega}}\end{pmatrix} :=\begin{pmatrix}\tilde{u}\\
\tilde{w}\end{pmatrix}=\tilde{\mathcal A_2}{\mathfrak u}\in
D(\tilde{\mathcal A_2})=D(\mathcal A_2),
$$
and by Theorem \ref{ident} there holds
$\tilde{u}|_{{\partial\Omega}}=\tilde w$. This
yields \eqref{went}.
\end{proof}

\begin{remark}\label{h3}
Take $u\in L^2(\Omega)$ such that
$\begin{pmatrix}u\\ u|_{{\partial\Omega}}\end{pmatrix}$ is in the domain
of $\tilde{\mathcal A}_2^2$. By the above corollary $u$ is a function in
$H^{3/2}(\Omega)$ such that the homogeneous boundary condition \eqref{went}
holds. Such boundary condition is expressed by means of a boundary
differential operator of order 2, hence by usual boundary regularity
results we obtain that $u\in H^3(\Omega)$. By induction one can in
fact see that
$$
C_c^\infty(\overline{\Omega})\times C^{\infty}(\partial\Omega)
\subset D(\tilde{\mathcal A}_2^\infty)\subset C^\infty(\overline{\Omega})
\times C^{\infty}(\partial\Omega).
$$
\end{remark}

Since the semigroup $\tilde{\mathcal T}_2$ generated by $\tilde{\mathcal A}_2$
is analytic, its smoothing effect yields
$$
\tilde{\mathcal T}_2(t)\left(\mathcal{X}^2\right)\subset
D(\tilde{\mathcal A}_2^\infty)\subset
\begin{cases}
D(\tilde{\mathcal A}_2^2)\\
C^\infty(\overline{\Omega})\times C^\infty(\partial\Omega)
\end{cases}
$$
for all $t>0$.

Thus, in view of Corollary \ref{wentz} and Remark \ref{h3} we finally
conclude that the initial value problem for the heat equation with
general Wentzell boundary conditions is well-posed.

\begin{corollary} \label{coro3.17}
Under the assumptions of Theorem \ref{ident}, let
$b\in \left(L^2(\Omega)\right)^n$, $c\in L^2(\Omega)$, and
$\tilde{\gamma}\in L^2(\partial\Omega)$. Then for all
$f\in L^2(\Omega)$ and all $g\in L^2(\partial\Omega)$ the
initial-boundary value problem
\begin{gather*}
\dot{u}(t,x) =\nabla\!\cdot\!(a\nabla u(t,x))+b(x)\!\cdot\!\nabla
u(t,x)+c(x)u(t,x),  \quad t>0,\; x\in\Omega,\\
\nabla\!\cdot\!\left(a(z)\nabla u(t,z)\right)+b(z)\!\cdot\!\nabla u(t,z)
+c(z)u(t,z)\\
+\langle a(z)\nabla u(t,z),\nu(z)\rangle +\tilde{\gamma}(z) u(t,z)=0,
\quad t>0,\; z\in\partial\Omega,\\
u(0,x)=f(x), \quad x\in \Omega,\\
u(0,z)=g(z), \quad z\in \partial\Omega,
\end{gather*}
admits a unique classical solution $u$, and $u(t,\cdot)$ is of class
$C^\infty$ for all $t>0$.
\end{corollary}

The well-posedness of the wave equation with general Wentzell
boundary conditions follows immediately from the self-adjointness
result of \cite{FGGGR05}, as explained in Goldstein's book
\cite{Go85}. We state it now for completeness. A similar result
in the one-dimensional case but on all $\mathcal{X}_p$ spaces,
$1\leq p<\infty$, has been obtained in \cite{Mu06}.

\begin{corollary}
Under the assumptions of Theorem \ref{ident}, let further
$b\in \left(L^2(\Omega)\right)^n$, $c\in L^2(\Omega)$, and
$\tilde{\gamma}\in L^2(\partial\Omega)$. Then for all
$f\in H^2(\Omega)$ and $g\in H^1(\Omega)$ the second order
initial-boundary value problem with dynamical boundary conditions
\begin{gather*}
\ddot{u}(t,x) =\nabla\!\cdot\!(a\nabla u(t,x))+b(x)\!\cdot\!\nabla u(x)
+c(x)u(x), \quad t\geq 0,\; x\in\Omega,\\
\ddot{u}(t,z) = -\langle a(z)\nabla u(t,z),\nu(z)\rangle
 -\tilde{\gamma}(z) u(t,z), \quad t\geq 0,\; z\in\partial\Omega,\\
u(0,x)=f(x), \quad \dot{u}(0,x)=g(x), \quad x\in \Omega.
\end{gather*}
admits a unique classical solution $u$. If further
 $f,g\in C^\infty_c(\overline{\Omega})$, then $u(t,\cdot)$ is of
 class $C^\infty$ for all $t\geq 0$, and in fact it satisfies the
general Wentzell boundary conditions
\begin{equation}\label{wentbc}
\begin{gathered}
\nabla\!\cdot\!\left(a(z)\nabla u(t,z)\right)+b(z)\!\cdot\!\nabla u(t,z)
+c(z)u(t,z)\\
 +\langle a(z)\nabla u(t,z),\nu(z)\rangle  +\tilde{\gamma}(z) u(t,z)=0,
\quad t\geq 0,\; z\in\partial\Omega.
\end{gathered}
\end{equation}
\end{corollary}

Let us finally identify the generators of the semigroups
${\mathcal T}_p$ on $\mathcal{X}^p$. We thus answer a question
that was adressed in \cite[\S~7.6]{Go04}.

\begin{theorem}\label{identp}
Let $\partial\Omega\in C^\infty$ and $a_{ij}\in C^\infty(\overline{\Omega})$,
$1\leq i,j\leq n$. Then for all $p\in [1,\infty]$ the generator
$\mathcal{A}_p$ of the semigroup $\mathcal{T}_p$ is given by
\begin{gather*}
\begin{aligned}
D(\mathcal{A}_p)=\Big\{&\begin{pmatrix} u\\ w\end{pmatrix}\in
W^{2-\frac{1}{p},p}(\Omega)\times W^{\frac{3}{2}-\frac{1}{p},p}(\partial\Omega):
\\
& w=u|_{{\partial\Omega}}\hbox{ and }\nabla\!\cdot\!(a\nabla u)
\in L^p(\Omega) \Big\},
\end{aligned}
\\
\mathcal{A}_p=\begin{pmatrix} \nabla\!\cdot\!(a\nabla) & 0\\
-\langle a\nabla,\nu\rangle  & -\gamma I\end{pmatrix}.
\end{gather*}
\end{theorem}

\begin{proof}
Let us prove the claim for $p>2$. We have already remarked that
$\mathcal{X}^p\hookrightarrow \mathcal{X}^q$ for all
$1\leq q\leq p\leq\infty$. Moreover, it follows by the
ultracontractivity of $\mathcal{T}_2$ that $\mathcal{X}^p$ is invariant
under $\mathcal{T}_2(t)$ for all $p>2$ and $t>0$.
Thus, by \cite[Prop.~II.2.3]{EN00} the generator of $\mathcal{T}_p$
 is the part of $\mathcal{A}_2$ in $\mathcal{X}^p$. The claimed
expressions of $\mathcal{A}_p$ and $D(\mathcal{A}_p)$ then follow
as consequences of usual results on traces, cf. \cite[Thm.~7.53]{Ad75}.

Take now some $p$ with $1\leq p<2$. By \cite[Thm.~1.4.1]{Da90}
one has that the adjoint semigroup of $({\mathcal T}_p(t))_{t\geq 0}$
on $\mathcal{X}_p$ is actually $({\mathcal T}_q(t))_{t\geq 0}$
on $\mathcal{X}_q$, where $p^{-1}+q^{-1}=1$. Set
\begin{align*}
{\mathcal D}_p=\Big\{&\begin{pmatrix} u\\ w\end{pmatrix}\in
W^{2-\frac{1}{p},p}(\Omega)\times W^{\frac{3}{2}-\frac{1}{p},p}
(\partial\Omega):\\
& w=u|_{{\partial\Omega}}\hbox{ and }\nabla\!\cdot\!(a\nabla u)
\in L^p(\Omega) \Big\}.
\end{align*}
Consider the operator ${\mathcal A}_p$ whose action on ${\mathcal D}_p$
is given by
$$
\mathcal{A}_p{\mathfrak u}=\begin{pmatrix} \nabla\!\cdot\!(a\nabla)u\\
-\langle a\nabla u,\nu\rangle  -\gamma u\end{pmatrix}.
$$
Reasoning as in the proof of Theorem \ref{ident} one can see that its
adjoint is actually $\mathcal{A}_q$, $p^{-1}+q^{-1}=1$.
Then, since the generator of the pre-adjoint semigroup
$(\mathcal{T}_p(t))_{t\geq 0}$ on $\mathcal{X}_p$ of
$(\mathcal{T}_q(t))_{t\geq 0}$ on $\mathcal{X}_q$ is the pre-adjoint
operator $\mathcal{A}_p$ of $\mathcal{A}_q$ it follows that $\mathcal{A}_p$
 with domain $D(\mathcal{A}_p)=\mathcal{D}_p$ generates the
$C_0$-semigroup $(\mathcal{T}_p(t))_{t\geq 0}$ on $\mathcal{X}_p$,
and the claim follows.
\end{proof}

\begin{remark}\label{mark} \rm
A semigroup $T$ on a Banach lattice $X$ is called~\emph{Markovian} if it
is real, positive, and $T(t){1} ={1}$ for all $t\geq 0$.
 One thus sees that a semigroup is Markovian if and only if it is real,
 positive, and its generator $A$ satisfies $A{1}=0$. It follows
from Theorem \ref{identp} that for all
$p\in[1,\infty]$, $\mathcal{A}_p {1}=0$ if and only if
$\gamma\equiv0$.
\end{remark}

\begin{corollary}\label{gamma2}
Fix $p\in [1,\infty)$. Under the assumptions of Theorem \ref{identp},
let further $b\in \left(L^\infty(\Omega)\right)^n$,
$c\in L^\infty(\Omega)$, and $\tilde{\gamma}\in L^\infty(\partial\Omega)$.
Then the operator $\tilde{\mathcal A}_p$ defined by
\begin{equation*}
\tilde{\mathcal A}_p:=\begin{pmatrix} \nabla\!\cdot\!(a\nabla)
+b\!\cdot\!\nabla +cI & 0\\
-\langle a\nabla,\nu\rangle  & -\tilde{\gamma} I\end{pmatrix}
\end{equation*}
with domain $D(\tilde{\mathcal A}_p):=D({\mathcal A}_p)$ generates an
 analytic semigroup on $\mathcal{X}^p$.
\end{corollary}

\begin{proof}
Write $\tilde{\mathcal A}_p$ as
$$
\tilde{\mathcal A}_p={\mathcal A}_p+\mathcal{B}_p,
$$
where
\begin{equation*}
{\mathcal B}_p:=\begin{pmatrix} b\!\cdot\!\nabla +cI & 0\\
0 & (\gamma-\tilde{\gamma})I\end{pmatrix}.
\end{equation*}
Observe that by assumption $b\!\cdot\!\nabla +cI$ and
$(\gamma-\tilde{\gamma})I$ are compact operators from
$W^{1+\epsilon,p}(\Omega)$ to $L^p(\Omega)$ and from
$W^{\epsilon,p}(\partial\Omega)$ to $L^p(\partial\Omega)$,
respectively, for all $\epsilon>0$. Hence it is clear that
${\mathcal B}_p$ is a relatively ${\mathcal A}_p$-compact perturbation,
and the claim follows by \cite[Cor.~2.17]{EN00}.
\end{proof}

\begin{remark}\label{nonau} \rm
Our approach based on positive forms and sub-Markovian
semigroups also allows to tackle some nonautonomous Cauchy
problems, at least in the one-dimensional case. In fact, we have
shown in Corollary \ref{comp} and Theorem \ref{identp} that the
operator matrix ${\mathcal A}_p$ generates for all
$p\in[1,\infty)$ a strongly continuous semigroup of contractions.
This essentially follows from the Assumptions \ref{basic} on the
coefficients $a$ and $\gamma$. If we allow for more general,
time-dependent coefficients, we are led to introduce a family of
operators on $\mathcal{X}^p$ defined by
\begin{equation*}
\mathcal{A}_p(t):=\begin{pmatrix} \nabla\!\cdot\!(a(t)\nabla) & 0\\
-\langle a(t)\nabla,\nu\rangle  & -\gamma(t) I\end{pmatrix}, \quad t\geq s,
\end{equation*}
for fixed $s\in\mathbb R$, with joint domain
$D(\mathcal{A}_p(t)):=D(\mathcal{A}_p)$.
We restrict ourselves to the one-dimensional case and, instead of the
Assumptions \ref{basic}, we impose the following.
\begin{enumerate}
\item $a(t)$ is a real valued $C^\infty[0,1]$-function such that
  $c_1\leq a(t,x)\leq C_1$ holds for suitable constants $0<c_1,C_1$,
   and all $t\geq s$, $x\in [0,1]$.
\item $\gamma_1(t),\gamma_2(t)$ are real numbers such that
   $0\leq \gamma_i(t)\leq C_2$ for a suitable constant $C_2$ and each
   $t\geq s$, $i=1,2$.
\item The mapping $t\mapsto a(t,\cdot)$ is continuously differentiable.
\item The mappings $t\mapsto \gamma_i(t)$ are continuously differentiable,
   $i=1,2$.
\end{enumerate}
We can now consider the temporally inhomogeneous problem
\begin{gather*}
\dot{u}(t,x) =(a u')'(t,x), \quad t\geq s,\; x\in(0,1),\\
\dot{u}(t,j) =(-1)^{j+1}a(t,j) u'(t,j)-\gamma_j(t) u(t,j)=0, \quad
t\geq s,\; j=1,2,\\
u(s,x)=f(x),  \quad x\in [0,1],
\end{gather*}
and rewrite it in an abstract form as
\begin{equation}\label{nACP}
\begin{gathered}
\dot{\mathfrak u}(t)={\mathcal A}_p(t){\mathfrak u}(t), \quad t\geq s,\\
{\mathfrak u}(s)={\mathfrak f}\in \mathcal{X}^p,\\
\end{gathered}
\end{equation}
for some $s\geq 0$. Due to the above assumptions on the coefficients,
all the operators ${\mathcal A}_p(t)$, $t\geq 0$, have joint domains
and further the family $({\mathcal A}_p(t))_{t\geq 0}$ is stable in
the sense of \cite[Def.~4.3.1]{Ta79}. We can thus apply the Kato-Tanabe
theory of hyperbolic nonautonomous problems and, by virtue of the
results in \cite[\S~4.4]{Ta79}, deduce well-posedness
(in a suitable sense) for $(\rm{nACP})$. We refer to \cite{Ka53}
and \cite{Ta79} for more details on the theory of fundamental
solutions to nonautonomous Cauchy problems. Asymptotic issues could
also be investigated (e.g., by means of the methods recently surveyed
in \cite{Sc04}), but this goes beyond the scope of this paper.
\end{remark}

\begin{proposition} \label{prop3.23}
Let $\Omega$ be connected. Then the positive semigroup ${\mathcal
T}_p$ on $\mathcal{X}_p$, $p\in [1,\infty)$, is irreducible.
\end{proposition}

\begin{proof}
Since the irreducibility of ${\mathcal T}_2$ is inherited by all
semigroups ${\mathcal T}_p$ on $\mathcal{X}_p,\, 1\leq p<\infty$
(a consequence of Corollary \ref{comp} and
\cite[Theorem 7.2.2]{Ar04}), we only consider the case $p=2$ and show that
\cite[Cor.~2.11]{Ou04} applies. Indeed, by Lemma \ref{main} and
Remark \ref{cont}, the form $\mathcal Q$ is densely defined,
positive, continuous, closed, and local.

We have to prove that if $U$ is an open subset of
$\Omega \times \partial \Omega$ such that
\begin{equation}\label{irr}
\left(\chi_U\cdot {\mathfrak f}\right)\in {\mathcal V}\quad
\hbox{for all } {\mathfrak f}\in{\mathcal V},
\end{equation}
then either $\mu(U)=0$ or $\mu\big((\Omega\times \partial\Omega)\setminus U\big)
=0$. Here $\mu$ denotes the measure introduced at the beginning of Section~2,
i.e., the direct sum of the Lebesgue measure $\lambda$ on $\Omega$ and the
 Hausdorff measure $\sigma$ on $\partial \Omega$.

Take then $U=V\times W$ open subset of $\Omega\times\partial\Omega$ such
that \eqref{irr} holds. By definition
\begin{equation}\label{decomp}
\chi_U=
\begin{pmatrix}
\chi_V\\
\chi_W
\end{pmatrix}.
\end{equation}
Now $V$ is an open subset of $\Omega$ and by \eqref{irr} we deduce that
\begin{equation*}
\left(\chi_V\cdot f\right)\in H^1(\Omega)\quad \hbox{for all }
f\in H^1(\Omega).
\end{equation*}
Recall now that the Laplace operator with Neumann boundary conditions
(whose associated form has domain $H^1(\Omega)$) is irreducible,
 cf. \cite[Thm.~4.4.5]{Ou04}. Hence, again by \cite[Cor.~2.11]{Ou04}
 we deduce that $\lambda(V)=0$ or $\lambda(\Omega\setminus V)=0$.
Since $V$ is an open set, $\lambda(V)=0$ can only happen if $V$ is empty,
so that also $U=\emptyset$ and $\mu(U)=0$.

Let us now consider the case of $\lambda(\Omega\setminus V)=0$. Observe
that the constant function ${\mathfrak f}=1$ belongs to $\mathcal V$,
so that by \eqref{irr},  \eqref{decomp}, and the definition of $\mathcal V$
we deduce that ${\chi}_V\in H^1(\Omega)$ and ${\chi}_W$ is its trace
on $\partial \Omega$. On the other hand, since $\lambda(\Omega\setminus V)=0$
 one sees that the characteristic function $\chi_V$ is identically 1 a.e.
in $\Omega$. Consequently, the trace of $\chi_V$ is identically 1 $\sigma$-a.e.
on $\partial\Omega$. Summing up, ${\chi}_W$ turns out to be
identically 1 $\sigma$-a.e. on $\partial\Omega$, so that $W=\partial \Omega$
up to a set of $\sigma$-measure zero. We conclude that the claim follows
since $\mu\big((\Omega\times \partial\Omega)\setminus U\big)
=\lambda(\Omega\setminus V)+ \sigma(\partial \Omega\setminus W)=0$.
\end{proof}

Due to the irreducibility of the semigroup, we can then apply known results
(see, e.g., \cite[\S~3.5.1]{Ar04}) and draw the following conclusion.

\begin{corollary}
Let $\Omega$ be connected. Then there exists a strictly positive rank-1
projection $\mathcal P$ such that
$$
\Vert e^{-s({\mathcal A})}{\mathcal T}_p(t)-{\mathcal P}\Vert
\le Me^{-\varepsilon t}, \quad t\geq 0,
$$
for some constant $M\geq 0$, $\epsilon>0$.
\end{corollary}

If in particular the coefficient $\gamma\equiv 0$, then we have seen
in Remark \ref{mark} that 0 is the spectral bound of all generators
${\mathcal A}_p$, $p\in[1,\infty)$, and by
 \cite[C-IV.2.10 and C-III.3.5.(d)]{Na86} we obtain the following.

\begin{corollary}
Let $\Omega$ be connected and $\gamma\equiv 0$. Then for the
semigroup ${\mathcal T}_p$ on $\mathcal{X}_p$, $p\in[1,\infty)$
the following assertions hold.
\begin{enumerate}
\item The limit ${\mathcal P}{\mathfrak f}
   :=\lim_{t\to\infty}{\mathcal T}_p(t){\mathfrak f}$ exists for every
   ${\mathfrak f}\in \mathcal{X}_p$.
\item ${\mathcal P}$ is a strictly positive projection onto
 $\ker {\mathcal A}$, the one-dimensional subspace of $\mathcal{X}_p$
spanned by the constant 1 function ${\chi}$.
\item There exists $M \geq 1$ such that
$$
\Vert {\mathcal T}_p(t)-{\mathcal P}\Vert\le Me^{\lambda_2 t},
\quad t\geq 0,
$$
where $\lambda_2$ is the largest nonzero eigenvalue of the generator $A$.
\end{enumerate}
 \end{corollary}

Again, we stress that the second largest eigenvalue of $\mathcal A$ can be
 explicitly computed in some concrete cases, cf. \cite[\S~9]{KMN03},
thus obtaining an estimate for the semigroup's rate of convergence (in norm!)
toward a projection.

More results on the asymptotic behaviour of the semigroups ${\mathcal T}_p$
will be obtained in the next section.

\section{The case $\gamma\not\equiv0$}

Throughout this section we impose the following conditions.

\begin{assum}\label{basic3}
The coefficient $\gamma$ does not identically vanish on the boundary of each connected component of $\Omega$.
\end{assum}

Under the Assumption \ref{basic3}, the properties of the cosine operator
function and of the semigroups associated with $\mathcal{Q}$ are
 essentially improved. The main reason is the following.

\begin{lemma}\label{inv}
Under the Assumptions \ref{basic} and \ref{basic3}, the operator
$\mathcal{A}_p$ is invertible for all $p\in[1,\infty]$.
\end{lemma}

\begin{proof}
Assume without loss of generality that $\Omega$ is connected, and
let $\mathfrak{u}=\begin{pmatrix}u\\ u|_{{\partial\Omega}}\end{pmatrix}
\in D(\mathcal{A}_2)$ such that $\mathcal{A}_2\mathfrak{u}=0$.
We obtain by definition of $\mathcal{A}_2$ that
$\sigma(\partial\Omega\setminus\partial V)>0$.
\begin{align*}
0&=\mathcal{Q}(\mathfrak{u},\mathfrak{u})\\
&=\int_\Omega (a \nabla u)\!\cdot\!\nabla u dx
  + \int_{\partial\Omega} \gamma\vert u\vert^2 d\sigma\\
&\geq  c_1 \int_\Omega \vert \nabla u\vert^2 dx
  + \int_{\partial\Omega} \gamma\vert u\vert^2 d\sigma.
\end{align*}
It follows by the assumptions on $\gamma$ that $u\in H^1(\Omega)$
is constant in $\Omega$ and vanishes somewhere on $\partial\Omega$.
Hence, $\mathfrak{u}\equiv 0$, and 0 is not an eigenvalue of $\mathcal{A}_2$.
Since $\mathcal{T}_2$ is compact and analytic, $\mathcal{A}_2$ has
compact resolvent. Therefore $\mathcal{A}_2$ is invertible, and the same
holds for all $\mathcal{A}_p$, since
$\sigma(\mathcal{A}_p)=\sigma(\mathcal{A}_2)$ for all $p\in[1,\infty]$.
\end{proof}

We already know by Remark \ref{integral}.(2) that the growth bound
of $\mathcal{T}_p$ is given by the spectral bound $s(\mathcal{A}_2)$.
By the above lemma such a spectral bound is strictly negative, and we
obtain the following.

\begin{corollary}\label{stab}
Under the Assumptions \ref{basic} and \ref{basic3}, the semigroup
$\mathcal{T}_p$ is uniformly exponentially stable for all $p\in[1,\infty]$.
\end{corollary}

\begin{corollary}\label{ap}
Under the Assumptions \ref{basic} and \ref{basic3}, the cosine operator
function generated by $\mathcal{A}_2$ is contractive, together with the
associated sine operator function. Moreover, the solutions to the second
order abstract Cauchy problem
\begin{equation}\label{ACP2}
\begin{gathered}
\ddot{\mathfrak u}(t)={\mathcal A}_2{\mathfrak u}(t), \quad t\in\mathbb{R},\\
{\mathfrak u}(0)={\mathfrak f}\in {\mathcal V},\\
\dot{\mathfrak u}(0)={\mathfrak g}\in{\mathcal{X}^2},\\
\end{gathered}
\end{equation}
are almost periodic.
\end{corollary}

\begin{proof}
By Theorem \ref{main1} and Lemma \ref{inv}, the operator $\mathcal{A}_2$
is self-adjoint and strictly negative definite. It follows by
\cite[Lemma~3.1]{Go69} that the reduction matrix associated with
${\mathcal A}_2$ generates a group of isometries, hence both the cosine
operator function $({\mathcal C}(t))_{t\in\mathbb{R}}$ generated by
$\mathcal{A}_2$ and the associated sine operator function
$({\mathcal S}(t))_{t\in\mathbb{R}}$ are contractive. More precisely,
by \eqref{groupcos} the classical solutions to \eqref{ACP2} are given by
$$
\mathfrak{u}(t)={\mathcal C}(t)\mathfrak{f}+{\mathcal S}(t)\mathfrak{g},
\quad t\in\mathbb{R}.
$$
Moreover, $\mathcal{A}_2$ has compact resolvent by Corollary \ref{comp},
hence by Lemma \ref{cofproposition} the solutions to \eqref{ACP2}
are almost periodic.
\end{proof}

Even more can be said if we replace the Assumption \ref{basic3} by
the following stronger version.

\begin{assum}\label{basic2} \rm
The coefficient $\gamma$ is strictly positive, i.e., there holds
 $c_2\leq \gamma$ $d\sigma$-a.e. for some constant $c_2>0$.
\end{assum}

We can now sharpen Lemma \ref{ultralemma} and obtain the following, cf.
also \cite[Prop.~2.6]{AMPR03}.

\begin{proposition}\label{ultralemma2}
Under the Assumptions \ref{basic} and \ref{basic2} the semigroup
$\mathcal{T}_2$ on $\mathcal{X}^2$ associated with $\mathcal Q$
satisfies the estimate
\begin{equation}\label{ultra3}
\Vert{\mathcal T}_2(t){\mathfrak f}\Vert_{\mathcal{X}^\infty} \leq
M_\mu t^{-\frac{\mu}{4}}\Vert{\mathfrak f}\Vert_{\mathcal{X}^2}
\quad\hbox{ for all }t>0,\; {\mathfrak f}\in{\mathcal{X}^2},
\end{equation}
for
\begin{equation*}
\mu\in
\begin{cases}
[2n-2,\infty), &\hbox{if } n\geq 3,\\
(2,\infty), &\hbox{if } n=2,\\
[1,\infty), &\hbox{if } n=1,\\
\end{cases}
\end{equation*}
and some constant $M_\mu$.
\end{proposition}

\begin{proof}
Take $\mathfrak{u}=\begin{pmatrix}u\\ u|_{{\partial\Omega}}\end{pmatrix}
\in\mathcal{V}$.
Observe that plugging \eqref{maz} into \eqref{sobolev} and \eqref{necas}
one obtains
\begin{gather}\label{sobolev2}
\Vert u\Vert_{L^\frac{2\mu}{\mu-2}(\Omega)}
\leq N_1 \left(\Vert \nabla u\Vert_{L^2(\Omega)}
+ \Vert u\Vert_{L^2(\partial\Omega)}\right),
\\ \label{necas2}
\Vert u\Vert_{L^\frac{2\mu}{\mu-2}(\partial\Omega)}\leq N_2 \left(\Vert \nabla u\Vert_{L^2(\Omega)} + \Vert u\Vert_{L^2(\partial\Omega)}\right)
\end{gather}
for suitable constants $N_1,N_2$, where $\mu\in [2n-2,\infty)$ if $n\geq 3$,
and $\mu\in (2,\infty)$ if $n\leq 2$. On the other hand there holds
$$
\Vert \nabla u\Vert^2_{L^2(\Omega)} + \Vert u\Vert^2_{L^2(\partial\Omega)}
\leq N_3 \mathcal{Q}(\mathfrak{u},\mathfrak{u}),
$$
where $N_3:=(c_1\vee c_2)^{-1}$. The claim then follows by
\cite[Thm.~2.4.2]{Da90} for $n\geq 2$, and by \cite[Thm.~2.4.6]{Da90}
for $n=1$.
\end{proof}

Combining the uniform exponential stability and the ultracontractivity of
$\mathcal{T}_2$ we finally derive the following $L^2-L^\infty$ stability
estimate.

\begin{corollary}\label{ultralemma3}
Under the Assumptions \ref{basic} and \ref{basic2} the semigroup
$\mathcal{T}_2$ on $\mathcal{X}^2$ associated with $ \mathcal Q$
satisfies the estimate
\begin{equation*}\label{ultra4}
\Vert{\mathcal T}_2(t){\mathfrak f}\Vert_{\mathcal{X}^\infty} \leq
M_\mu
\big(\frac{1-ts(\mathcal{A}_2)}{t}\big)^{\mu/4}
e^{ts(\mathcal{A}_2)}\Vert{\mathfrak f}\Vert_{\mathcal{X}^2}
\quad\hbox{for all }t>0,\; {\mathfrak f}\in{\mathcal{X}^2},
\end{equation*}
where $\mu,M_\mu$ are as in Proposition \ref{ultralemma2}.
\end{corollary}

\begin{proof}
The claim is a direct consequence of Remark \ref{integral}.(2),
Proposition \ref{ultralemma2}, and \cite[Lemma~6.5]{Ou04}.
\end{proof}

We can reformulate the above results aas follows.

\begin{corollary}
The semigroup $\mathcal{T}_2$ satisfies the estimate
\begin{equation}\label{ultra6}
\Vert{\mathcal T}_2(t){\mathfrak f}\Vert_{\mathcal{X}^\infty} \leq
e^{\kappa(t)}\Vert{\mathfrak f}\Vert_{\mathcal{X}^2} \quad\hbox{
for all }t>0,\; {\mathfrak f}\in{\mathcal{X}^2},
\end{equation}
where $\kappa$ is a function related to the estimate \eqref{ultra3}
and such that
\begin{equation}\label{kappa}
\kappa(\varepsilon)\sim C-\frac{n-1}{2}\log\varepsilon\quad\hbox{
as }\varepsilon\to 0^+
\end{equation}
for some constant $C>0$, if $n\geq 3$.
\end{corollary}

\begin{proof}
By Proposition \ref{ultralemma2}, it follows that the estimate
 \eqref{ultra6} holds with
$$
\kappa(t):=\log M_\mu - \frac{\mu}{4}\log t.
$$
for all $\mu\in [2n-2,\infty)$ if $n\geq 3$, $\mu\in (2,\infty)$
if $n=2$, or $\mu\in[1,\infty)$ if $n=1$. If in particular $n\geq 3$,
then \eqref{kappa} holds for some constant $C>0$.
\end{proof}

Note that this is a special case of the nonlinear result in
\cite{FGGR05} specialized to the linear case. In \cite{FGGR05}
it is shown that an estimate analogous to the above one holds with
$\Vert f\Vert_{\mathcal{X}_2}$ replaced by $\Vert
f\Vert_{\mathcal{X}_1}$, which is a much stronger result.

A direct computation shows that
for $\mu$ in the ranges defined above $\kappa$ is a continuous, monotonically decreasing function on $(0,\infty)$. We thus apply \cite[Thm.~2.2.3]{Da90} and finally derive the logarithmic Sobolev inequality
\begin{equation}\label{logar}
\int_{\overline{\Omega}}\mathfrak{f}^2\log \mathfrak{f}d\mu\leq
\varepsilon\mathcal{Q}(\mathfrak{f},\mathfrak{f})+\kappa(\varepsilon)\Vert
\mathfrak{f}\Vert^2_{\mathcal{X}^2}+ \Vert
\mathfrak{f}\Vert^2_{\mathcal{X}^2}\log \Vert
\mathfrak{f}\Vert_{\mathcal{X}^2},
\end{equation}
which is valid for all
$0\leq \mathfrak{f}\in \mathcal{V}\cap \mathcal{X}^\infty$ and all
 $\epsilon>0$.

\begin{remark}\label{varie} \rm
(1) We have already seen in Remark \ref{integral}.1)
 that $\mathcal{T}_2$ has a bounded, positive integral kernel.
If we assume $\gamma$ to be strictly positive, we can derive from
Corollary \ref{ultra4} the alternative estimate
$$
{\mathcal K}(t,{\bf x},{\bf y})\leq M_\mu^2
\left(\frac{1-ts(\mathcal{A}_2)}{t}\right)^{\frac{\mu}{2}}e^{2ts
(\mathcal{A}_2)}
\quad \hbox{ for all }t> 0,\;\hbox{ a.e. } {\bf x},{\bf
y}\in\overline{\Omega},
$$
 on the upper bound of the integral kernel, cf. \cite[\S~2.1]{Da90}.
Here $\mu$ and $M_\mu$ are the same parameters that appear in
Proposition \ref{ultralemma2}.

(2) The logarithmic Sobolev inequality \eqref{logar} for $\kappa$ of
the form $\kappa(\varepsilon) = C -\frac{n}{4}\log\varepsilon$ is typical
for uniformly elliptic operators with Dirichlet boundary conditions on
connected domains of $\mathbb{R}^n$, cf. \cite[\S~2.3]{Da90}.
Now, our $\mathcal{A}_2$ may be regarded as a differential operator
on $\overline{\Omega}$, where $\Omega$ is an $n$-dimensional bounded open
domain and $\partial\Omega$ is an $(n-1)$-dimensional manifold.
The results of \cite{FGGR05} give the best estimate of the form
\eqref{ultra6} near $t=0$; in fact, they agree with the best estimate
for the linear heat equation.
\end{remark}

Davies has developed (see \cite[\S~3.2]{Da90} and references therein)
 a method that makes use of such logarithmic Sobolev inequalities
for sesquilinear forms and allows to deduce that the associated
semigroups admit Gaussian estimates
\emph{with respect to a suitable metric}, cf. \cite[Thm.~3.2.7]{Da90}.
In view of Remark \ref{varie}.(2), this means in our context that a
 certain mild form of domination of $\mathcal{T}_2$ by the Gaussian
semigroup (in $\mathbb{R}^{2n-2}$, if $n\geq 2$, or in $\mathbb R$,
if $n=1$) holds -- where, again, the metric we endow $\mathbb{R}^{2n-2}$
or $\mathbb R$ with is a suitable one that needs not be equivalent to the
 Euclidean metric.

We point out that Gaussian estimates are a key argument for discussing several
issues, including the $p$-independence of the angle of analyticity of the
semigroups on $L^p$, $L^1$-analyticity, and the boundedness of the
$H^\infty$-calculus of their generators in $L^p$. We refer the reader
to \cite[\S~7.1]{Ou04}, \cite[\S~7.4]{Ar04}, and references therein.

\section{Appendix: A remainder of the theory of cosine operator functions}

We summarize a few generalities from the theory of cosine operator
functions as presented, e.g. in \cite{Fa85} or \cite[\S~3.14]{ABHN01}.

\begin{definition}\label{cof}
Let $X$ be a Banach space. A strongly continuous function
$C:{\mathbb R}\to {\mathcal L}(X)$ is called a {\em cosine operator function}
if it satisfies the D'Alembert functional relations
\begin{gather*}
C(t+s)+C(t-s)=2C(t)C(s),\quad t,s\in\mathbb R,\\
C(0)=I_X.
\end{gather*}
Further, the operator $A$ on $X$ defined by
$$
Ax:=\lim_{t\to 0}\frac{2}{t^2}(C(t)x-x),\quad
D(A):=\big\{x\in X: \lim_{t\to
0}\frac{2}{t^2}(C(t)x-x)\;\hbox{exists}\big\},
$$
is called the {\em generator of} $(C(t))_{t\in\mathbb R}$. We define the
associated {\em sine operator function} $(S(t))_{t\in\mathbb R}$
by
$$
S(t)x:=\int_0^t C(s)x ds,\quad t\in{\mathbb R},\; x\in X.
$$
\end{definition}

\begin{lemma}\label{wellp2char}
Let $A$ be a closed operator on a Banach space $X$. Then the operator
$A$ generates a cosine operator function $(C(t))_{t\in\mathbb R}$ on $X$,
with associated sine operator function $(S(t))_{t\in\mathbb R}$,
if and only if there exists a Banach space $V$, with
$[D(A)]\hookrightarrow V\hookrightarrow X$, such that the operator matrix
\begin{equation*}
{\mathbf A}:=\begin{pmatrix}
0 & I_V\\
A & 0
\end{pmatrix},\quad D({\mathbf A}):=D(A)\times V,
\end{equation*}
generates a $C_0$-semigroup $(e^{t\mathbf A})_{t\geq 0}$ in $V\times X$, and in this case there holds
\begin{equation}\label{groupcos}
e^{t{\mathbf A}}=\begin{pmatrix}
C(t) & S(t)\\
AS(t) & C(t)
\end{pmatrix},\quad t\geq 0.
\end{equation}
If such a space $V$ exists, then it is unique. The (unique) product
space ${\bf X}=V\times X$ is called \emph{phase space associated with}
$(C(t))_{t\in{\mathbb R}}$ (or \emph{with $A$}).
\end{lemma}

\begin{lemma}\label{pert}
Let $A$ generate a cosine operator function with associated phase space $V\times X$. Then also $A+B$ generates a cosine operator function with associated phase space $V\times X$, provided $B$ is an operator that is bounded from $V$ to $X$.
\end{lemma}

Concerning regularity, it is known that cosine operator functions have in general no smoothing effect (see \cite[Prop.~4.1]{TW77}). However, the following can be deduced by \eqref{groupcos} and the fact that a semigroup leaves invariant the domains of all of its generator's powers.

\begin{lemma}\label{regul}
Let $A$ generate a cosine operator function $(C(t))_{t\in\mathbb R}$
with associated sine operator function $(S(t))_{t\in\mathbb R}$.
Consider the solution to the second order abstract Cauchy problem
\begin{gather*}
\ddot{u}(t)=Au(t),\quad t\in\mathbb{R},\\
u(0)=f,\quad\dot{u}(0)= g,
\end{gather*}
which is given by $u(t)=C(t)f+S(t)g$, $t\in{\mathbb R}$. Then $u(t)\in D(A^k)$
for all $t\in\mathbb R$, provided that $f,g\in D(A^{2k})$, $k\in\mathbb N$.
\end{lemma}

It is known that cosine and sine operator functions cannot be stable
-- i.e., one cannot expect the decay of the norm of a solution to
a second order abstract Cauchy problem. Hence, one is usually
interested in boundedness and almost periodicity of such solutions.
The following results are due to Arendt and Batty, cf. \cite[Cor.~5.6]{AB97}.

\begin{lemma}\label{cofproposition}
Let $A$ generate a bounded cosine operator function $(C(t))_{t\in\mathbb R}$
with associated sine operator function $(S(t))_{t\in\mathbb R}$ on a Banach
space $X$. If $A$ has compact resolvent, then $(C(t))_{t\in\mathbb R}$ is
almost periodic. If further $A$ is invertible, then also
$(S(t))_{t\in\mathbb R}$ is almost periodic.
\end{lemma}

\subsection*{A technical lemma}

\begin{lemma}\label{lemmadense}
Let $X_1,X_2,Y_1,Y_2$ be Banach spaces, such that $X_1\subset X_2$,
and $Y_1$ is dense in $Y_2$. Consider a surjective operator
$L\in{\mathcal L}(X_1,Y_1)$ such that $\ker(L)$ is dense in $X_2$. Then
$$
\left\{\begin{pmatrix}x\\ y\end{pmatrix}\in X_1\times Y_1: Lx=y\right\}
$$
is dense in $X_2\times Y_2$.
\end{lemma}

\begin{proof}
Let $x\in X_2$, $y\in Y_2$, $\epsilon>0$. Take $z\in Y_1$ such that
$\Vert y-z\Vert_{Y_2}<\epsilon$. The surjectivity of $L$ ensures that
there exists $u\in X_1$ such that $Lu=z$. Take $\tilde u,\tilde x\in \ker(L)$
such that $\Vert u-\tilde u\Vert_{X_2}<\epsilon$ and
$\Vert x-\tilde x\Vert_{X_2}<\epsilon$. Let $w:=\tilde x+u-\tilde u\in X_1$.
Then
\begin{align*}
&\big\Vert
\begin{pmatrix}x\\y\end{pmatrix}
- \begin{pmatrix}w\\z\end{pmatrix}
\big\Vert_{X_2\times Y_2}\\
&\leq\big\Vert
\begin{pmatrix}
x-\tilde x\\ 0
\end{pmatrix}
\big\Vert_{X_2\times Y_2}
+ \big\Vert
\begin{pmatrix}
u-\tilde u\\ 0
\end{pmatrix}
\big\Vert_{X_2\times Y_2}
+\big\Vert
\begin{pmatrix}0\\ y- z
\end{pmatrix}\big\Vert_{X_2\times Y_2}
< 3\epsilon.
\end{align*}
Since $L(w)=L(u)=z$, we obtain
$\begin{pmatrix}w\\z\end{pmatrix}\in X_1\times Y_1$.
\end{proof}

\subsection*{Acknowledgments}
The authors warmly thank W. Arendt and E. Ouhabaz for the opportunity
to read \cite{Ar04} and \cite{Ou04} in a preprint form,
thus inspiring most of the ideas and techniques appearing in this paper.
Moreover, the authors express their gratitude to G.R. Goldstein and
J.A. Goldstein for careful reading and valuable comments to this paper.

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\end{document}