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\AtBeginDocument{{\noindent\small Nonlinear Differential Equations,\newline
{\em Electron. J. Diff. Eqns.},
Conf. 05, 2000, pp. 51--67.\newline
http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline
ftp ejde.math.swt.edu \quad ejde.math.unt.edu (login: ftp)}
\thanks{\copyright 2000 Southwest Texas State University.}\vspace{1cm}}
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\begin{document}
\setcounter{page}{51}
\title[Existence and perturbation of principal eigenvalues]
{Existence and perturbation of\\ principal eigenvalues for a\\
periodic-parabolic problem}
\author[Daniel Daners] {Daniel Daners}
\address{Daniel Daners \newline\indent
Department of Mathematics\\
Brigham Young University\\
Provo, Utah 84602, USA}
\curraddr{School of Mathematics and Statistics\\
Univ. of Sydney\\
NSW~2006\\
Australia}
\email{D.Daners@maths.usyd.edu.au}
\dedicatory{Dedicated to Alan Lazer on his 60th birthday}
\thanks{Published October 24, 2000.}
\subjclass{35K20, 35P05, 35B20, 47N20}
\keywords{principal eigenvalues, periodic-parabolic problems,
\newline\indent
parabolic boundary-value problems, domain perturbation.}
\begin{abstract}
We give a necessary and sufficient condition for the existence of a
positive principal eigenvalue for a periodic-parabolic problem with
indefinite weight function. The condition was originally established
by Beltramo and Hess [\textit{\frenchspacing Comm. Part. Diff.
Eq.}, \textbf{9} (1984), 919--941] in the framework of the
Schauder theory of classical solutions. In the present paper, the
problem is considered in the framework of variational evolution
equations on arbitrary bounded domains, assuming that the
coefficients of the operator and the weight function are only
bounded and measurable. We also establish a general perturbation
theorem for the principal eigenvalue, which in particular allows
quite singular perturbations of the domain. Motivation for the
problem comes from population dynamics taking into account seasonal
effects.
\end{abstract}
\maketitle
\section{Introduction}
Population models with diffusion taking into account seasonal effects
are often described by a periodic-parabolic problem. The habitat of
the population is represented by a bounded domain
$\Omega\subset\mathbb R^N$ ($N=2$ or $3$ in a real model), and the
diffusion by an elliptic operator, $\mathcal A(t)$, having time
periodic coefficients of period $T>0$ (the length of one cycle). The
linearization of such a boundary value problem at a periodic solution
leads to a periodic-parabolic eigenvalue problem of the form
\begin{equation}
\label{eq:evpm}
\begin{aligned}
\partial_t u+\mathcal A(t) u
&=\lambda m u&&\text{in }\Omega\times[0,T],\\
u(\cdot\,,t)
&=0&&\text{on }\partial\Omega\times [0,T],\\
u(\cdot,0) &= u(\cdot,T)&\qquad&\text{in }\Omega,
\end{aligned}
\end{equation}
with weight function $m$. It is of particular importance to know the
existence of a positive principal eigenvalue of \eqref{eq:evpm},
which, by definition, is a number $\lambda$ such that \eqref{eq:evpm}
has a nontrivial nonnegative solution. The notion of a principal
eigenvalue for periodic-parabolic problems was introduced and
motivated in Lazer~\cite{lazer:82:rps} (see also Castro \&
Lazer~\cite{castro:82:rps}). More applications can for instance be
found in Hess~\cite{hess:91:ppb}.
\par
In this paper we prove two results. First, we establish a necessary
and sufficient condition on the weight function $m$ which guarantees
the existence of a positive principal eigenvalue of \eqref{eq:evpm}.
Second, we provide a general perturbation result for the eigenvalues
of \eqref{eq:evpm} allowing quite singular perturbations of the domain
$\Omega$. All results will be proved in the framework of weak
solutions. This requires the principal part of $\mathcal A(t)$ to be
in divergence from, but allows us to deal with arbitrary domains
$\Omega$, and only requires the coefficients of $\mathcal A(t)$ and
the weight function $m$ to be bounded and measurable. Note that, as a
special case, the results apply to weighted elliptic eigenvalue
problems (c.f.~\cite[Remark~16.5]{hess:91:ppb}).
\par
Working in the framework of the Schauder theory of classical solutions
Beltramo \& Hess \cite{beltramo:84:pep} (see also
\cite{beltramo:84:hpp,hess:91:ppb}) found necessary and sufficient
conditions for the existence of a positive principal eigenvalue. It
was somewhat a surprise that, unlike in case of the corresponding
elliptic problem, it is not sufficient that $m$ be positive somewhere
in $\Omega$. The relevant condition turned out to be
\begin{displaymath}
\mathcal P(m):=\frac{1}{T}\int_0^T\sup_{x\in\Omega}m(x,t)\,dt>0.
\end{displaymath}
We will show that a similar result holds under our assumptions. As the
weight function $m$ is only assumed to be bounded and measurable, we
will need to replace the supremum by the essential supremum. The
problem was also considered in Daners~\cite{daners:97:ppe}, where, in
addition to the hypotheses in the present paper, it was assumed that
$m$ is lower semi-continuous. Godoy, Lami~Dozo \&
Paczka~\cite{godoy:97:ppe} were able to deal with bounded and
measurable weight functions $m$. However they kept the smoothness
assumptions on the coefficients of $\mathcal A(t)$ and the domain made
in the original theorem by Beltramo and Hess. They moreover required
the top order coefficients of $\mathcal A(t)$ to be continuously
differentiable. The reason was that in the proof they needed to
rewrite $\mathcal A(t)$ in divergence form. We find it more natural to
assume from the beginning that the operator be in divergence form, and
then to get rid of the smoothness assumptions all together.
\par
We then prove two perturbation results. The first asserts that any
finite set of eigenvalues of \eqref{eq:evpm} is upper semi-continuous
with respect to the domain, the coefficients of $\mathcal A(t)$, and
the weight function $m$. The second determines the behaviour of the
principal eigenvalue of a sequence of approximating problems. It turns
out that the limit exists, and is the smallest positive principal
eigenvalue. The perturbation theorems improve and complement similar
results in Daners~\cite{daners:97:ppe}. We relax the conditions on the
domain convergence, and not necessarily assume that the limiting set
$\Omega$ be connected.
\par
An outline of the paper is as follows. In Section~\ref{sec:main} we
give the precise assumptions and state our main results. In
Section~\ref{sec:mainsteps} we discuss the main steps of the proof of
the existence result. In Section~\ref{sec:pert} we prove our general
perturbation results. The techniques introduced there also give rise
to an approximation procedure, which allows to pass from results known
in the smooth case to the non-smooth case. This procedure is described
and exploited in Section~\ref{sec:approx}. In
Section~\ref{sec:upper-est} we prove some spectral estimates providing
the key to establish the existence of a positive principal eigenvalue.
We close the paper by two appendices, the first outlining the changes
necessary in \cite{daners:96:dpl} to relax the notion of domain
convergence, and the second to prove a technical result about convex
functions.
\section{Assumptions and Main Results}
\label{sec:main}
Throughout let $\Omega\subset\mathbb R^N$ be a bounded open set, and
let $T$ be a fixed positive number. Moreover suppose that $\mathcal
A(t)$ satisfies the following assumptions.
%
\begin{assumption}
\label{ass:A}
Suppose that $\mathcal A(t)$ is defined by
\begin{equation}
\label{eq:A}
\mathcal A(t)u
:=-\sum_{i=1}^N\frac{\partial}{\partial x_i}
\Bigl(
\sum_{j=1}^Na_{ij}(\cdot,t)\frac{\partial}{\partial x_j}u
\Bigr)
+\sum_{i=1}^N b_i(\cdot,t)\frac{\partial}{\partial x_i}u
+c_0(\cdot,t)u,
\end{equation}
where $a_{ij}=a_{ji}, b_i, c_0\in L_\infty(\Omega\times (0,T))$.
Moreover we assume that there exists $\alpha>0$, called the
ellipticity constant, such that
\begin{equation}
\label{eq:Aelliptic}
\sum_{i=1}^N\sum_{j=1}^Na_{ij}(x,t)\xi_i\xi_j\geq\alpha|\xi|^2
\end{equation}
for all $(x,t)\in\Omega\times(0,T)$ and $\xi\in\mathbb R^N$.
\end{assumption}
%
By a solution of \eqref{eq:evpm} we always mean a weak solution (for a
definition see e.g.~\cite{trudinger:68:peq}). It is well known that
weak solutions are classical solutions if the domain, the coefficients
of $\mathcal A(t)$ and the weight function are smooth enough. The set
of $\lambda\in\mathbb C$ such that
\begin{displaymath}
\partial_t u+\mathcal A(t) u-\lambda m u=f
\end{displaymath}
in $\Omega\times[0,T]$ subject to the boundary conditions in
\eqref{eq:evpm} has a bounded inverse on $L_2(\Omega\times(0,T))$ is
called the \textit{resolvent set} of \eqref{eq:evpm}. The complement
of the resolvent set is called the spectrum of \eqref{eq:evpm}. We
call $\lambda$ a [\textit{principal}] \textit{eigenvalue} of
\eqref{eq:evpm} if \eqref{eq:evpm} has a nontrivial [nonnegative]
solution. Such a nontrivial solution is said to be a
[\textit{principal}] \textit{eigenfunction} of \eqref{eq:evpm} to the
[\textit{principal}] \textit{eigenvalue} $\lambda$.
%
\subsection*{Existence of a positive principal eigenvalue}
We next state our main result on the existence of a positive principal
eigenvalue of \eqref{eq:evpm}. We define
\begin{equation}
\label{eq:P(m)}
\mathcal P(m):=\frac{1}{T}\int_0^T\essup_{x\in\Omega}m(x,t)\,dt>0.
\end{equation}
If $\Omega$ is a bounded domain (an open and connected set) we have
the following theorem. The assertions are wrong in general if $\Omega$
is not connected (c.f.~Remark~\ref{rem:notcon}).
%
\begin{theorem}
\label{thm:exist}
Suppose that $\Omega$ is a bounded domain, that $\mathcal A(t)$ is
as above with $c_0\geq 0$, and that $m\in
L_\infty(\Omega\times(0,T))$. Then the following assertions are
equivalent:
\begin{enumerate}
\item $\mathcal P(m)>0$.
\item Problem \eqref{eq:evpm} has a positive principal eigenvalue.
\item Problem \eqref{eq:evpm} has an eigenvalue with positive real
part.
\end{enumerate}
In this case the positive principal eigenvalue, $\lambda_1$, is the
only principal eigenvalue with positive real part, and
\begin{equation}
\label{eq:char}
\lambda_1=\inf\{\realpart\lambda\colon\text{$\lambda$ is an
eigenvalue of \eqref{eq:evpm} with $\realpart\lambda>0$}\}.
\end{equation}
\end{theorem}
%
\begin{remark}
Note that the above theorem can also be used to give necessary and
sufficient conditions for the existence of a negative principal
eigenvalue of \eqref{eq:evpm}. We only need to replace $m$ by $-m$.
\end{remark}
%
\subsection*{Perturbation of the spectrum}
To state our perturbation results we need some additional definitions
and assumptions. We first look at domain convergence
(c.f.~\cite[Section~2]{dancer:96:rcp}).
%
\begin{definition}
\label{def:domconv}
Suppose that $\Omega$ is a bounded open set (not necessarily
connected), and that $\Omega_n$ are bounded domains (connected by
definition). We say that $\Omega_n$ converges to $\Omega$, in
symbols $\Omega_n\to\Omega$, if
\begin{itemize}
\item[(i)] $\displaystyle\lim_{n\to\infty}
\meas(\Omega_n\cap\bar\Omega^\complement)=0$.
\item[(ii)] There exists a compact set $K\subset\Omega$ of capacity
zero such that for each compact set $\Omega'\subset\Omega\setminus
K$ there exists $n_0\in\mathbb N$ such that $\Omega'\subset
\Omega_n$ for all $n\geq n_0$.
\end{itemize}
\end{definition}
%
\begin{remark}
\label{rem:unbounded}
In the above definition we did not assume that $\Omega_n$
stays in the same bounded subset of $\mathbb R^N$. In fact the
diameter of $\Omega_n$ may tend to infinity as long as the measure
of $\Omega_n\cap\Omega^\complement$ converges to zero.
\end{remark}
%
As usual we denote by $W_2^1(\Omega)$ the standard Sobolev space, and
by $\ocirc W_2^1(\Omega)$ the closure of the set of all smooth
functions with compact support in $W_2^1(\Omega)$.
%
\begin{definition}
\label{def:stable}
An open set $\Omega\subset \mathbb R^N$ is said to be
\textit{stable} if for each $u\in W_2^1(\mathbb R^N)$ with support
in $\bar\Omega$ we have that $u\in\ocirc W_2^1(\Omega)$.
\end{definition}
%
The stability of an open set is a very weak regularity condition. It
can be characterized by means of capacities (see e.g.~Adams \&
Hedberg~\cite[Theorem~11.4.1]{adams:96:fsp}). Next we state our
assumptions on the perturbed operators $\mathcal A_n(t)$. As we assume
that the coefficients are only bounded and measurable we can always
extend them to $\mathbb R^N$ in such a way that the ellipticity
constant remains unchanged.
%
\begin{assumption}
\label{ass:A_n}
For all $n\in\mathbb N$ let $\mathcal A_n(t)$ be an operator of the
form \eqref{eq:A} with coefficients $a_{ij}^{(n)}=a_{ji}^{(n)},
b_i^{(n)}, c_0^{(n)}\in L_\infty(\mathbb R^N\times (0,T))$. Suppose that
\begin{displaymath}
\sup_{\substack{i,j=1,\dots,N\\ n\in\mathbb N}}
\bigl\{
\|a_{ij}^{(n)}\|_\infty,\|b_i^{(n)}\|_\infty,\|c_0^{(n)}\|_\infty
\bigr\}
<\infty,
\end{displaymath}
and that $a_{ij}^{(n)}, b_i^{(n)}$ and $c_0^{(n)}$ converge to the
corresponding coefficients of $\mathcal A(t)$ in
$L_{2,\mathrm{loc}}(\mathbb R^N\times(0,T))$. Finally suppose that
the sequence of ellipticity constants of $\mathcal A_n(t)$ has a
positive lower bound.
\end{assumption}
%
We finally consider the weight functions. Note that we can assume them
to be defined on $\mathbb R^N$ by simply extending them by zero
outside $\Omega$.
%
\begin{assumption}
\label{ass:m_n}
Let $m_n,m\in L_\infty(\mathbb R^N\times(0,T))$ for all $n\in\mathbb
N$, assume that $\|m_n\|_\infty$ is a bounded sequence, and that
$m_n$ converges to $m$ in $L_{2,\mathrm{loc}}(\mathbb
R^N\times(0,T))$.
\end{assumption}
%
We are now in a position to state our main perturbation results. What
we mean by the multiplicity of an eigenvalue of \eqref{eq:evpm} we
explain in Definition~\ref{def:mult}.
%
\begin{theorem}
\label{thm:evpm-pert}
Suppose that $c_0\geq 0$, and that Assumption~\ref{ass:A_n}
and~\ref{ass:m_n} are satisfied. Further assume that
$\Omega\subset\mathbb R^N$ is a stable bounded open set, and that
$\Omega_n\to\Omega$ in the sense of Definition~\ref{def:domconv}.
Finally let $U\subset\mathbb C$ be an open set containing exactly
$r$ eigenvalues of \eqref{eq:evpm}. Then, counting multiplicity,
the perturbed problem
\begin{equation}
\label{eq:evpm_n}
\begin{aligned}
\partial_t u+\mathcal A_n(t) u
&=\lambda m_n u&&\text{in }\Omega_n\times[0,T]\\
u(\cdot\,,t)
&=0&&\text{on }\partial\Omega_n\times [0,T]\\
u(\cdot,0) &= u(\cdot,T)&\qquad&\text{in }\Omega_n
\end{aligned}
\end{equation}
has exactly $r$ eigenvalues in $U$ for $n\in\mathbb N$ sufficiently
large.
\end{theorem}
%
The proof of the above theorem is given in Section~\ref{sec:pert} (it
follows from Proposition~\ref{prop:ppm} and
Theorem~\ref{thm:ppm-pert}). The next theorem determines what happens
to a sequence of positive principal eigenvalues if we pass to the
limit. The main problem is that $\Omega$ is not assumed to be
connected, and thus the limiting problem might have more than one
positive principal eigenvalue.
%
\begin{theorem}
\label{thm:pev-pert}
Suppose the assumptions of Theorem~\ref{thm:evpm-pert} hold, and
that \eqref{eq:evpm} admits a positive principal eigenvalue. Then
for all $n\in\mathbb N$ large enough \eqref{eq:evpm_n} has a unique
positive principal eigenvalue $\lambda_n$. The sequence
$(\lambda_n)$ converges to a positive principal eigenvalue of
\eqref{eq:evpm}, and this eigenvalue can be characterized by
\eqref{eq:char}.
\end{theorem}
%
The above is a consequence of Theorem~\ref{thm:evpm-pert} and some
spectral estimates. The proof is given in Lemma~\ref{lem:pev-conv}
and~\ref{lem:pev-exist}.
%
\begin{remark}
\label{rem:stable}
If $\Omega_n\subset\Omega$ for all $n\in\mathbb N$ then the above
results remains true without assuming that $\Omega$ is stable
(c.f.~\cite[Remark~3.2(a)]{daners:96:dpl}).
\end{remark}
%
\begin{remark}
\label{rem:non-unique}
If $\Omega$ is not connected the spectrum of \eqref{eq:evpm} is the
union of the spectra of the corresponding problems on the components
of $\Omega$. Hence, the limiting problem may have several principal
eigenvalues, or one with higher algebraic multiplicity.
\end{remark}
%
\begin{remark}
\label{rem:notcon}
If $\Omega$ is not connected it is possible for \eqref{eq:evpm} not
to have a positive principal eigenvalue even though $\mathcal
P(m)>0$. As an example look at a domain with two connected
components, $\Omega_1$ and $\Omega_2$. Then the spectrum of
\eqref{eq:evpm} is the union of the spectra of the corresponding
problems on $\Omega_1$ and $\Omega_2$. If we set
$m_i:=m|_{\Omega_i}$ ($i=1,2$), then one can easily arrange that
$\mathcal P(m_i)\leq 0$ for $i=1,2$, but $\mathcal P(m)>0$. The
reason is that the location where the essential supremum of $m$
occurs may shift from $\Omega_1$ to $\Omega_2$ as $t$ increases from
$0$ to $T$. Suppose that we are in this situation, and that
$\Omega_n$ are domains approximating $\Omega$ in the sense of
Definition~\ref{def:domconv} (for instance connect $\Omega_1$ and
$\Omega_2$ by a small strip shrinking to a line). If $m_n$ is the
weight function on $\Omega_n$ then $\mathcal P(m_n)>0$ for large
$n\in\mathbb N$, and, by Theorem~\ref{thm:exist}, there exists a
positive principal eigenvalue, $\lambda_n$, for the perturbed
domain. However, as the limiting problem does not have a principal
eigenvalue, and $0$ is not an eigenvalue, $\lambda_n$ must converge
to infinity as $n$ goes to infinity. In fact, the upper bound of
$\lambda_n$ established in Lemma~\ref{lem:pev-exist} also goes to
infinity. The reason is that the curve $\gamma$ and the function
$\varphi_0$ used there cannot be chosen the same for all
$n\in\mathbb N$.
\end{remark}
%
\begin{remark}
In Theorem~\ref{thm:pev-pert} it can be shown that, if normalized to
one in the space ${L_2(\Omega\times(0,T))}$, at least a subsequence of the
eigenfunctions converges to an eigenfunction of the limiting problem
in ${L_2(\Omega\times(0,T))}$ (see proof of
Lemma~\ref{lem:pev-conv}). However, if $\lambda_0$ is of higher
multiplicity we cannot expect the whole sequence to converge. For
the convergence of eigenfunctions see also
Daners~\cite[Theorem~3.2]{daners:97:ppe}.
\end{remark}
%
\begin{remark}
Note that, as a special case, our perturbation results can be
applied to weighted elliptic boundary value problems of the from
\begin{displaymath}
\begin{aligned}
\mathcal Au&=\lambda mu&\qquad&\text{in $\Omega$},\\
u&=0&&\text{on $\partial\Omega$}
\end{aligned}
\end{displaymath}
(c.f.~\cite[Remark~16.5]{hess:91:ppb}). Domain perturbations of
weighted elliptic eigenvalue problems were also considered in
L\'opez-G\'omez~\cite{lopez:96:mpe}.
\end{remark}
\section{Main Steps of the Existence Proof}
\label{sec:mainsteps}
In this section we outline the main steps of the proof of
Theorem~\ref{thm:exist}. The basic idea, which was already exploited
by Beltramo \& Hess \cite{beltramo:84:pep}, is to look at the family
of auxiliary eigenvalue problems
\begin{equation}
\label{eq:evpaux}
\begin{aligned}
\partial_t u+\mathcal A(t) u-\lambda mu
&=\mu u&&\text{in }\Omega\times[0,T],\\
u(\cdot\,,t)
&=0&&\text{on }\partial\Omega\times [0,T],\\
u(\cdot,0) &= u(\cdot,T)&\qquad&\text{in }\Omega,
\end{aligned}
\end{equation}
where the parameter $\lambda$ ranges over $\mathbb R$. We throughout
assume that $m$ is a bounded and measurable function on
$\Omega\times[0,T]$. Concerning the existence of a principal
eigenvalue for \eqref{eq:evpaux} the following is known (see
\cite[Section~2]{daners:97:ppe}).
%
\begin{lemma}
\label{lem:aux}
For each $\lambda\in\mathbb R$ the eigenvalue problem
\eqref{eq:evpaux} has a unique principal eigenvalue. This eigenvalue
is real, algebraically simple, and the corresponding eigenfunction
can be chosen to be continuous and positive in $\Omega\times[0,T]$.
\end{lemma}
%
The continuity of the eigenfunction follows from the regularity theory
for weak solutions of parabolic equations, the positivity follows from
the periodicity and the weak Harnack inequality for parabolic
equations (e.g.~\cite{trudinger:68:peq}).
\par
For every $\lambda\in\mathbb R$ denote the principal eigenvalue of
\eqref{eq:evpaux} by $\mu(\lambda)$. Note that $\lambda$ is a
principal eigenvalue of \eqref{eq:evpm} if and only if
$\mu(\lambda)=0$. Hence, to prove Theorem~\ref{thm:exist} we need
criteria ensuring that $\mu(\cdot)$ has a unique positive zero.
The properties of $\mu(\cdot)$ leading to this conclusion are
summarized in the following proposition.
%
\begin{proposition}
\label{prop:mu}
The function $\mu(\cdot)$ has the following properties:
\begin{enumerate}
\item\label{mu-concave} $\mu(\cdot)\colon\mathbb R\to\mathbb R$ is
concave.
\item\label{mu0pos} If $c_0\geq 0$ then $\mu(0)>0$.
\item\label{mu-infty} $\lim_{\lambda\to\infty}\mu(\lambda)=-\infty$ if
and only if $\mathcal P(m)>0$.
\item\label{mu-neg} If $\lambda\in\mathbb C$ is an eigenvalue of
\eqref{eq:evpm} with $\realpart\lambda>0$ then
$\mu(\realpart\lambda)\leq 0$.
\end{enumerate}
\end{proposition}
%
The above proposition can be used as follows to prove
Theorem~\ref{thm:exist}.
%
\begin{proof}[Proof of Theorem~\ref{thm:exist}]
Assuming that $c_0\geq 0$ we have from (\ref{mu0pos}) that
$\mu(0)>0$. By (\ref{mu-concave}) the function $\mu(\cdot)$ is
concave and hence continuous. Thus, by (\ref{mu-infty}), the first
two assertions of Theorem~\ref{thm:exist} are equivalent. Next, due
to (\ref{mu-neg}) and (\ref{mu-infty}), the first assertion of
Theorem~\ref{thm:exist} is equivalent to the third one. Finally, the
uniqueness of a positive principal eigenvalue of \eqref{eq:evpm}
follows from the concavity of $\mu(\cdot)$. The characterization
\eqref{eq:char} is a consequence of (\ref{mu-neg}). This completes
the proof of Theorem~\ref{thm:exist}.
\end{proof}
%
It remains to prove Proposition~\ref{prop:mu}. The first two
properties, (\ref{mu-concave}) and (\ref{mu0pos}), are established in
\cite{daners:97:ppe}, the first as part of the proof of Theorem~2.1 on
p.~391, and the second in Lemma~2.4. The proof of (\ref{mu-neg}) will
be given in Lemma~\ref{lem:mu-neg} using the result in the smooth case
and an approximation procedure. It remains to prove (\ref{mu-infty}).
The necessity of the condition $\mathcal P(m)>0$ clearly follows from the
lower estimate
\begin{equation}
\label{eq:lower-est}
\mu(\lambda)\geq \mu(0)-\lambda\mathcal P(m)
\end{equation}
valid for all $\lambda\geq 0$. A proof is given in
Lemma~\ref{lem:lower-est}. The most difficult part is to show that
$\mathcal P(m)>0$ implies that
\begin{equation}
\label{eq:muneg}
\lim_{\lambda\to\infty}\mu(\lambda)=-\infty
\end{equation}
The proof of the above assertion is quite technical and requires an
upper estimate for $\mu(\lambda)$. To state the estimate in a concise
form we define
\begin{equation}
\label{eq:Ab}
A:=[a_{ij}]_{1\leq i,j\leq N}
\quad\text{and}\quad
b:=[b_1,\dots,b_N]^\tp.
\end{equation}
Note that due to the ellipticity condition \eqref{eq:Aelliptic} the
matrix $A(x,t)$ is invertible for almost all
$(x,t)\in\Omega\times(0,T)$. Let $\mathcal D(\Omega)$ denote the set
of smooth functions with compact support in $\Omega$. Finally, denote
the support of a function $u$ by $\supp u$. The following result is an
obvious consequence of Proposition~\ref{prop:upper-est}.
%
\begin{proposition}
\label{prop:m-upper-est}
Suppose that $\gamma\in C^1(\mathbb R,\mathbb R^N)$ is $T$-periodic.
Further assume that $\varphi_0\in\mathcal D(\Omega)$ is a
nonnegative function such that
\begin{equation}
\label{eq:intone}
T\int_\Omega \varphi_0^2\,dx=1,
\end{equation}
and suppose that $\varphi(x,t):=\varphi_0(x-\gamma(t))
\in\Omega\times\mathbb R$ for all $(x,t)\in
\supp(\varphi_0)\times\mathbb R$. Furthermore, let
$w:=\varphi(b-d\gamma/dt)+2A(\nabla \varphi)^\tp$. Then, for all
$\lambda\in\mathbb R$
\begin{equation}
\label{eq:m-upper-est}
\mu(\lambda)\leq
\frac{1}{4}\int_0^T\int_\Omega w^\tp A^{-1}w
+\varphi^2c_0\,dx\,dt
-\lambda\int_0^T\int_\Omega \varphi^2m\,dx\,dt.
\end{equation}
\end{proposition}
%
Our claim \eqref{eq:muneg} follows from
Proposition~\ref{prop:m-upper-est} if $\gamma$ and $\varphi_0$ can be
chosen such that
\begin{equation}
\label{eq:intpos}
\int_0^T\int_{\Omega}
[\varphi_0(x-\gamma(t))]^2m(x,t)\,dx\,dt>0.
\end{equation}
The idea is that $\mathcal P(m)>0$ implies that the integral of $m$
over a tubular neighbourhood about a periodic curve is positive.
Godoy, et.~al~\cite[Lemma~4.4]{godoy:97:ppe} showed that there exists
a $T$-periodic curve $\gamma\in C^1(\mathbb R,\mathbb R^N)$ and an
open set $\Omega_0\subset\Omega$ with the property that
$x-\gamma(t)\in\Omega$ for all $(x,t)\in\bar\Omega_0\times[0,T]$, and
\begin{displaymath}
\int_0^T\int_{\Omega_0} m(x-\gamma(t),t)\,dx\,dt>0.
\end{displaymath}
Choosing an appropriate function $\varphi_0\in\mathcal D(\Omega_0)$
normalized by \eqref{eq:intone} we easily get the following lemma.
%
\begin{lemma}
\label{lem:intpos}
If $\mathcal P(m)>0$, then in Proposition~\ref{prop:m-upper-est} the
curve $\gamma$ and the function $\varphi_0$ can be chosen such that
\eqref{eq:intpos} holds.
\end{lemma}
%
The above lemma together with \eqref{eq:m-upper-est} shows that
$\mathcal P(m)>0$ implies \eqref{eq:muneg} and thus completes the
proof of Proposition~\ref{prop:mu}.
\section{Perturbation Results}
\label{sec:pert}
The main purpose of this section is to prove
Theorem~\ref{thm:evpm-pert} and~\ref{thm:pev-pert}. We start by
studying the periodic-parabolic problem
\begin{equation}
\label{eq:pp}
\begin{aligned}
\frac{\partial}{\partial t}u+\mathcal A(t)u+\mu u&=f
&\qquad&\text{in }\Omega\times(0,T),\\
u&=0&&\text{on }\partial\Omega\times(0,T),\\
u(\cdot,0)&=u(\cdot,T)&&\text{in }\Omega.
\end{aligned}
\end{equation}
It can be shown that, for each $\mu\in\mathbb R$ large enough, the
above problem has unique weak solution
\begin{displaymath}
u\in L_2((0,T),\ocirc W_2^1(\Omega))\cap C([0,T],L_2(\Omega))
\end{displaymath}
for all $f\in L_2((0,T),W_2^{-1}(\Omega))$ (see
\cite[Theorem~2.2]{daners:96:dpl} or
\cite[Theorem~3.6.1]{lions:72:nbv}). Define the resolvent operator
$R_\mu$ by $R_\mu f:=u$ for all $f\in L_2((0,T),W_2^{-1}(\Omega))$.
Then for all $p\geq 2$
\begin{displaymath}
R_\mu\in\mathcal L\bigl(L_p(\Omega\times(0,T)\bigr)
\cap \mathcal L\bigl(C([0,T],L_2(\Omega))\bigr)
\end{displaymath}
is a compact operator (see \cite[Section~5]{daners:96:dpl}). Suppose
now that $p>N/2$, and that $f\in L_p((0,T)\Omega))$ is a nontrivial
nonnegative function. If $u$ is the corresponding solution of
\eqref{eq:pp} with $f\in L_p((0,T)\times\Omega))$ then the weak
Harnack inequality, the regularity theory for parabolic equations (see
e.g.~\cite{trudinger:68:peq}) and periodicity show that $u\in
C(\Omega\times[0,T])$, and $u(x,t)>0$ for all
$(x,t)\in\Omega\times[0,T]$. We next look at the perturbed
periodic-parabolic problem
\begin{equation}
\label{eq:pp_n}
\begin{aligned}
\frac{\partial}{\partial t}u+\mathcal A_n(t)u+\mu u&=f_n
&\qquad&\text{in }\Omega_n\times(0,T),\\
u&=0&&\text{on }\partial\Omega_n\times(0,T),\\
u(\cdot,0)&=u(\cdot,T)&&\text{in }\Omega_n.
\end{aligned}
\end{equation}
We suppose that $\mathcal A_n(t)$ satisfies Assumption~\ref{ass:A_n},
and that $\Omega_n\to\Omega$ in the sense of
Definition~\ref{def:domconv}. Further denote by $R_{\mu,n}$ the
resolvent operator of \eqref{eq:pp_n}.
%
\begin{theorem}
\label{thm:pp-pert}
Suppose that the above assumptions are satisfied, and that
$\mu\in\mathbb R$ is large enough. Then for all $p\leq 2<\infty$ the
resolvent $R_{\mu,n}$ converges to $R_\mu$ in ${\mathcal
L\bigl(L_p(\Omega\times(0,T)\bigr)}$. Moreover, if ${f_n
\rightharpoonup f}$ weakly in ${L_p(\mathbb R^N\times(0,T))}$,
then the solutions of \eqref{eq:pp_n} converge to the solution of
\eqref{eq:pp} strongly in ${L_p(\mathbb R^N\times(0,T))}$.
\end{theorem}
\begin{proof}
For a slightly weaker notion of domain convergence the above theorem
was proved in \cite[Theorem~5.1]{daners:96:dpl}. Note that all
results in that paper only depend on
\cite[Theorem~3.1]{daners:96:dpl}, so we only need to generalize
this theorem for our definition of domain convergence. The necessary
modifications of the proof are given in Appendix~\ref{app:dompert}.
\end{proof}
%
Suppose now that $M$ is the multiplication operator induced by $m\in
L_\infty(\Omega\times(0,T))$ on $L_2(\Omega\times(0,T))$. If $c_0\geq
0$ we know from Proposition~\ref{prop:mu}(\ref{mu0pos}) that $R:=R_0$
exists. Hence, taking into account the compactness of $R$, the
operator $R\circ M$ is compact on $L_2(\Omega\times(0,T))$. It easily
follows that $\lambda\in\mathbb C$ is in the spectrum of
\eqref{eq:evpm} if and only if $\lambda^{-1}$ is in the spectrum of
$R\circ M$. By the spectral theory for compact operators
(e.g.~\cite[Theorem~III.6.26]{kato:76:ptl}) all eigenvalues are of
finite algebraic multiplicity.
%
\begin{definition}
\label{def:mult}
By the multiplicity of an eigenvalue of \eqref{eq:evpm} we mean the
multiplicity of $\lambda^{-1}$ as an eigenvalue of $R\circ M$.
\end{definition}
%
The above reasoning leads to the following proposition.
%
\begin{proposition}
\label{prop:ppm}
The spectrum of \eqref{eq:evpm} consists of eigenvalues of finite
algebraic multiplicity. Moreover, $\lambda\in\mathbb C$ is an
eigenvalue of \eqref{eq:evpm} if and only if $\lambda^{-1}$ is an
eigenvalue of $R\circ M$.
\end{proposition}
%
We next look at perturbations of $R\circ M$. We we set $R_n:=R_{0,n}$,
and denote the multiplication operator induced by $m_n$ by $M_n$. The
following theorem is a reformulation and extension of
Theorem~\ref{thm:evpm-pert}.
%
\begin{theorem}
\label{thm:ppm-pert}
Suppose that $c_0\geq 0$, that $\mathcal A_n(t)$ and $m_n$ satisfy
Assumption~\ref{ass:A_n} and \ref{ass:m_n}, respectively, and that
$\Omega_n\to\Omega$ in the sense of Definition~\ref{def:domconv}.
Then for all $p\leq 2<\infty$ the operator $R_{n}\circ M_n$
converges to $R\circ M$ in ${\mathcal
L\bigl(L_p(\Omega\times(0,T)\bigr)}$. If $U\subset\mathbb C$ is an
open set containing exactly $r$ eigenvalues of $R\circ M$ then,
counting multiplicity, $U$ contains exactly $r$ eigenvalues of
$R_n\circ M_n$ for all $n$ sufficiently large.
\end{theorem}
\begin{proof}
The first assertion of the theorem is a simple consequence of
Theorem~\ref{thm:pp-pert} applying similar arguments as in
\cite[Theorem~5.1]{daners:96:dpl}. The second assertion follows from
the first by applying a general perturbation theorem
(Kato~\cite[Section~IV.3.5]{kato:76:ptl}).
\end{proof}
%
The remainder of this section is devoted to the proof of
Theorem~\ref{thm:pev-pert}. We first show that the limit of a sequence
of principal eigenvalues is a principal eigenvalue. The main
difficulty in the proof is that $\Omega$ is not assumed to be
connected.
%
\begin{lemma}
\label{lem:pev-conv}
For each $n\in\mathbb N$ let $\lambda_n$ be a principal eigenvalue
of \eqref{eq:evpm_n}, and assume that the sequence $(\lambda_n)$
converges to some $\lambda_1\in\mathbb R$. Then $\lambda_1$ is a
principal eigenvalue of \eqref{eq:evpm}. If $\lambda_1>0$ then it
can be characterized by \eqref{eq:char}.
\end{lemma}
\begin{proof}
Let $u_n$ denote an eigenfunction to the principal eigenvalue
$\lambda_n$ of \eqref{eq:evpm_n}, and assume that $\lambda_n$
converges to $\lambda_1$ as $n$ goes to infinity. We can assume that
$u_n>0$ in $\Omega\times(0,T)$, and normalize it in
${L_2(\Omega\times(0,T))}$ to norm one. Then, $(u_n)_{n\in\mathbb
N}$ is a bounded sequence in a Hilbert space, and therefore has a
weakly convergent subsequence $(u_{n_k})_{k\in\mathbb N}$ with limit
$u$ (e.g.~\cite[Section~V.2]{yosida:80:faa}). By our hypotheses on
$m_n$ (see Assumption~\ref{ass:m_n}) it follows that $\lambda_{n_k}
m_{n_k} u_{n_k}$ converges to $\lambda_1mu$ weakly in
${L_2(\Omega\times(0,T))}$. But then
\cite[Theorem~5.1]{daners:96:dpl} and the results in
Appendix~\ref{app:dompert} imply that $u_{n_k}$ converges to $u$
strongly in $L_2(\mathbb R^N\times(0,T))$. Hence $u$ is nontrivial
and nonnegative, proving that $\lambda_1$ is a principal eigenvalue
of \eqref{eq:evpm}.
\par
It remains to show that, if $\lambda_1>0$, then \eqref{eq:evpm} has
no eigenvalue with positive real part smaller than $\lambda_1$.
Suppose, to the contrary, that \eqref{eq:evpm} has an eigenvalue
$\nu\in\mathbb C$ with $0<\realpart\nu<\lambda_1$. Then, by
Theorem~\ref{thm:ppm-pert}, it follows that \eqref{eq:evpm_n} has an
eigenvalue $\mu_n\in\mathbb C$ with $0<\realpart\nu_n<\lambda_n$ for
all $n\in\mathbb N$ large enough. As $\Omega_n$ is connected
$\lambda_n$ can be characterized by \eqref{eq:char}, leading to a
contradiction. Hence, $\lambda_1$ is also given by \eqref{eq:char}.
\end{proof}
%
Theorem~\ref{thm:pev-pert} follows from the above lemma if we can show
the existence and convergence of a positive principal eigenvalue of
the perturbed problem \eqref{eq:evpm_n}.
%
\begin{lemma}
\label{lem:pev-exist}
Suppose the assumptions of Theorem~\ref{thm:pev-pert} hold. Then,
for $n$ sufficiently large, the perturbed eigenvalue problem
\eqref{eq:evpm_n} has a unique positive principal eigenvalue
converging to a positive principal eigenvalue of
\eqref{eq:evpm}.
\end{lemma}
\begin{proof}
We start by proving the existence of a positive principal eigenvalue
of the perturbed eigenvalue problem \eqref{eq:evpm_n}. To do so we
consider the family of auxiliary eigenvalue problems
\begin{equation}
\label{eq:evpaux_n}
\begin{aligned}
\partial_t u+\mathcal A_n(t) u-\lambda m_n u
&=\mu u&&\text{in }\Omega_n\times[0,T],\\
u(\cdot\,,t)
&=0&&\text{on }\partial\Omega_n\times [0,T],\\
u(\cdot,0) &= u(\cdot,T)&\qquad&\text{in }\Omega_n.
\end{aligned}
\end{equation}
By Lemma~\ref{lem:aux} the above eigenvalue problem has a unique
principal eigenvalue, $\mu_n(\lambda)$, for all $\lambda\in\mathbb
R$. We show that $\mu_n(0)$ is a bounded sequence. To do so fix a
function $\varphi\in\mathcal D(\Omega\setminus K)$, where $K$ is the
set from Definition~\ref{def:domconv} of domain convergence. By
Definition~\ref{def:domconv}(ii) the support of $\varphi$ is
contained in $\Omega_n$ if $n$ is sufficiently large. Applying
Proposition~\ref{prop:m-upper-est} we therefore have that
\begin{displaymath}
\mu_n(\lambda)\leq
\frac{1}{4}\int_0^T\int_{\Omega_n} w_n^\tp A_n^{-1}w_n
+\varphi^2c_0^{(n)}\,dx\,dt
\end{displaymath}
for all $n\in\mathbb N$ sufficiently large. Here $w_n$ and $A_n$ are
the expressions corresponding to $w$ and $A$ for the perturbed
problem. By our assumptions it is easy to see that the right hand
converges. Thus the sequence $\mu_n(\lambda)$ is bounded from above.
It is bounded from below as $\mu_n(0)>0$ for all $n\in\mathbb N$ by
Proposition~\ref{prop:mu}(\ref{mu0pos}). Hence, there exists a
subsequence $\bigl(\mu_{n_k}(0)\bigr)_{k\in\mathbb N}$ converging to
$\mu_0\geq 0$ as $k$ goes to infinity. By Lemma~\ref{lem:pev-conv}
the limit $\mu_0$ is a principal eigenvalue of \eqref{eq:evpaux}
with $\lambda=0$. Next we note that zero cannot be an eigenvalue of
\eqref{eq:evpm} as otherwise it would be an eigenvalue of
\eqref{eq:evpm} on a component of $\Omega$. Since we assumed that
$c_0\geq 0$ this is not possible by
Proposition~\ref{prop:mu}(\ref{mu0pos}). Hence $\mu_0>0$. By
Lemma~\ref{lem:pev-conv} the eigenvalue $\mu_0$ is characterized as
the one with the smallest positive real part. Hence, every
convergent subsequence of $\bigl(\mu_n(0)\bigr)_{n\in\mathbb N}$
tends to $\mu_0$ and thus the whole sequence converges. As $\mu_0>0$
we also have that $\mu_n(0)>0$ for $n\in\mathbb N$ large enough.
\par
Next we show that, for $n$ sufficiently large, \eqref{eq:evpm_n} has
a positive principal eigenvalue. We assumed in
Theorem~\ref{thm:pev-pert} that \eqref{eq:evpm} has a positive
principal eigenvalue. Note that $\Omega$ is not necessarily
connected, so the spectrum of \eqref{eq:evpm} is the union of the
spectra of the corresponding problems on the components. Hence we
can select a connected component $\Omega_1\subset\Omega$ such that
\eqref{eq:evpm} has a positive principal eigenvalue on $\Omega_1$.
By Theorem~\ref{thm:exist} it follows that $\mathcal
P(m|_{\Omega_1})>0$. Due to Lemma~\ref{lem:intpos} there exists a
$T$-periodic curve $\gamma\in C^1(\mathbb R,\mathbb R^N)$, a
function $\varphi_0\in\mathcal D(\Omega_1)$ satisfying
\eqref{eq:intone} such that \eqref{eq:intpos} holds. Setting
$\varphi(x,t) :=\varphi_0(x-\gamma(t))$ we see that
\begin{displaymath}
\lim_{n\to\infty}\int_0^T\int_{\Omega_n}\varphi^2m_n\,dx\,dt
=\int_0^T\int_{\Omega_n}\varphi^2m_n\,dx\,dt.
\end{displaymath}
As the right hand side of the above equation is positive there
exists $\delta>0$ and $n_0\in\mathbb N$ such that
\begin{equation}
\label{eq:intpos_n}
\int_0^T\int_{\Omega_n} \varphi^2m_n\,dx\,dt>\delta
\end{equation}
for all $n\geq n_0$. Next observe that by
Proposition~\ref{prop:m-upper-est}
\begin{equation}
\label{eq:m-upper-est_n}
\mu_n(\lambda)\leq
\frac{1}{4}
\int_0^T\int_{\Omega_1} w_n^\tp A_n^{-1}w_n
+\varphi^2c_0^{(n)}\,dx\,dt
-\lambda\int_0^T\int_{\Omega_n}\varphi^2m_n\,dx\,dt
\end{equation}
for all $n\in\mathbb N$ and all $\lambda\in\mathbb R$. For each
$n\geq n_0$ we thus have $\mu_n(\lambda)<0$ if only $\lambda$ is
large enough. We showed already that $\mu_n(0)>0$, so by
Proposition~\ref{prop:mu}(\ref{mu-concave}) the function
$\mu_n(\cdot)$ has a unique positive zero, $\lambda_n$, whenever
$n\in\mathbb N$ is large enough. This proves the existence of a
unique positive principal eigenvalue for \eqref{eq:evpm_n} if $n$ is
large.
\par
It remains to show that $\lambda_n$ converges to a principal
eigenvalue of \eqref{eq:evpm}. To do so we first establish a bound
on $\lambda_n$. From \eqref{eq:m-upper-est_n} and
\eqref{eq:intpos_n} we conclude that
\begin{displaymath}
\lambda_n\leq\frac{1}{4\delta}
\int_0^T\int_{\Omega_n} w_n^\tp A_n^{-1}w_n
+\varphi^2c_0^{(n)}\,dx\,dt
\end{displaymath}
for all $n\in\mathbb N$ large enough. It is easy to see that the
right hand side of the above inequality converges. Hence, the
sequence $(\lambda_n)_{n\in\mathbb N}$ is bounded from above. On the
other hand we know already that $\lambda_n>0$ for all $n\in\mathbb
N$. Thus the sequence $(\lambda_n)_{n\in\mathbb N}$ is bounded. We
next show that it converges. Due to the boundedness we can extract a
subsequence converging to some $\lambda_1\geq 0$. By
Lemma~\ref{lem:pev-conv} $\lambda_1$ is a principal eigenvalue of
\eqref{eq:evpm}. We already showed that zero is no principal
eigenvalue, so $\lambda_1>0$. Moreover, Lemma~\ref{lem:pev-conv}
asserts that $\lambda_1$ is the eigenvalue of \eqref{eq:evpm} with
the smallest positive real part. Hence all convergent subsequences
of $(\lambda_n)$ tend to $\lambda_1$, and thus the
whole sequence converges. This completes the proof of the lemma.
\end{proof}
\section{Approximation Procedures}
\label{sec:approx}
We now want to introduce an approximation procedure which allows to
pass from results known for smooth data to the case of non-smooth
data. The idea is to regularize $\mathcal A(t)$, $m$ and $\Omega$. We
start with $\mathcal A(t)$, and assume that it satisfies
Assumption~\ref{ass:A}. In a first step we extend the coefficients of
$\mathcal A(t)$ from $\Omega\times(0,T)$ periodically to
$\Omega\times\mathbb R$, and then extend its first and zero order
coefficients $b_i$ and $c_0$ by zero outside $\Omega\times\mathbb R$.
Next we extend $a_{ij}$ by $\alpha\delta_{ij}$ to $\mathbb R^{N+1}$,
where $\delta_{ij}$ is the Kronecker symbol and $\alpha>0$ the
ellipticity constant of $\mathcal A(t)$. In abuse of notation we
denote this new operator again by $\mathcal A(t)$. It has the same
ellipticity constant as the original one. We then fix nonnegative
functions $\varphi\in\mathcal D(\mathbb R^N)$ and $\psi\in\mathcal
D(\mathbb R)$ satisfying
\begin{equation}
\label{eq:normalize}
\int_{\mathbb R^N}\varphi(x)\,dx=1
\quad\text{and}\quad
\int_{-\infty}^{\infty}\psi(t)\,dt=1.
\end{equation}
For all $n\in\mathbb N$ define $\varphi_n$ and $\psi_n$ by
$\varphi_n(x):=n^N\varphi(nx)$ and $\psi_n(t):=n\varphi(nt)$,
respectively. Then $(\varphi_n)_{n\in\mathbb N}$ and
$(\psi_n)_{n\in\mathbb N}$ are mollifiers on $\mathbb R^N$ and
$\mathbb R$, respectively. Clearly
$\Phi_n(x,t):=\varphi_n(x)\psi_n(t)$ defines a mollifier on $\mathbb
R^{N+1}$. For all $n\in\mathbb N$ and $i,j=1,\dots,N$ we set
\begin{displaymath}
a_{ij}^{(n)}:=\Phi_n*a_{ij},\quad b_i^{(n)}:=\Phi_n*b_i
\quad\text{and}\quad
c_0^{(n)}:=\Phi_n*c_0,
\end{displaymath}
and define $\mathcal A_n(t)$ to be the operator of the form
\eqref{eq:A} with these coefficients. Using the definition of the
convolution and the properties of the mollifiers
(e.g.~\cite[Section~8.2]{folland:84:ran}) it is straightforward to
check that $\mathcal A_n(t)$ satisfies Assumption~\ref{ass:A_n}.
\par
We next look at the weight function $m\in
L_\infty(\Omega\times(0,T))$. We first extend it periodically to
$\Omega\times\mathbb R$, and then by $-\|m\|_\infty$ outside
$\Omega\times\mathbb R$. Then the approximations $m_n$ defined by
$m_n:=\Phi_n*m$ clearly satisfy Assumption~\ref{ass:m_n}.
\par
To approximate the bounded domain $\Omega$ let $\Omega_n$ be a
sequence of sub-domains of class $C^\infty$ exhausting $\Omega$. Then
$\Omega_n\to\Omega$ in the sense of Definition~\ref{def:domconv}. Note
that in this case we do not need to assume that $\Omega$ is stable in
order to apply the results from Section~\ref{sec:pert}
(c.f.~Remark~\ref{rem:stable}). Finally define
\begin{displaymath}
\mathcal P(m_n):=\frac{1}{T}\int_0^T\essup_{x\in\Omega_n}m(x,t)\,dt.
\end{displaymath}
We then have the following lemma.
%
\begin{lemma}
\label{lem:limP_n}
Under the above assumptions $\mathcal P(m_n)$ converges, and
\begin{equation}
\label{eq:limP_n}
\lim_{n\to\infty}\mathcal P(m_n)\leq\mathcal P(m).
\end{equation}
\end{lemma}
\begin{proof}
By the definition of $m_n$ we have that
\begin{displaymath}
\essup_{y\in\Omega_n}m_n(y,t)
\leq\int_{\mathbb R^N}\varphi_n(x-z)\int_{-\infty}^\infty\psi_n(t-s)
\essup_{y\in\Omega}m(y,s)\,ds\,dz.
\end{displaymath}
As we extended $m$ by $-\|m\|_\infty$ outside $\Omega\times\mathbb
R$ the essential supremum on the right hand side is the same as the
essential supremum over $\mathbb R^N$. Taking into account
\eqref{eq:normalize} the above inequality reduces to
\begin{displaymath}
\essup_{y\in\Omega_n}m_n(y,t)
\leq\psi_n*\essup_{y\in\Omega}m(y\,,\cdot)(t)
\end{displaymath}
for all $t\in[0,T]$. As $(\psi_n)_{n\in\mathbb N}$ is a mollifier
the right hand side of the above inequality converges to
$\essup_{y\in\Omega}m(y\,,\cdot)$ almost everywhere in $(0,T)$. As
all functions involved are bounded uniformly with respect to
$n\in\mathbb N$, an application of the dominated convergence theorem
yields \eqref{eq:limP_n}.
\end{proof}
%
\begin{remark}
It can also be shown that $P(m)\leq\liminf_{n\to\infty}P(m_n)$. The
proof is based on the trivial inequality
$m_n(x,t)\leq\essup_{x\in\Omega_n}m_n(x,t)$ and Fatou's lemma. The
above inequality is true for every sequence $m_n$ approaching $m$
pointwise almost everywhere, whereas \eqref{eq:limP_n} requires more
properties of $m_n$.
\end{remark}
%
For every $\lambda\in\mathbb R$ let $\mu_n(\lambda)$ and
$\mu(\lambda)$ denote the unique principal eigenvalues of
\eqref{eq:evpaux_n} and \eqref{eq:evpaux}, respectively.
%
\begin{proposition}
\label{prop:ev-conv}
Under the above assumptions $\mu_n(\lambda)$ converges to
$\mu(\lambda)$ uniformly with respect to $\lambda$ in bounded sets
of $\mathbb R$ as $n$ goes to infinity.
\end{proposition}
\begin{proof}
By Lemma~\ref{lem:aux} the eigenvalues $\mu_n(\lambda)$ and
$\mu(\lambda)$ are algebraically simple. Hence
Theorem~\ref{thm:ppm-pert} with $m=1$ implies that $\mu_n(\lambda)$
converges to $\mu(\lambda)$ for all $\lambda\in\mathbb R$. By
Proposition~\ref{prop:mu}(\ref{mu-concave}) the functions
$\mu_n\colon\mathbb R\to\mathbb R$ are concave, and thus by the
results in Appendix~\ref{app:convex} local uniform convergence on
$\mathbb R$ follows.
\end{proof}
%
Using the approximation procedure just introduced we next establish
the lower estimate \eqref{eq:lower-est} for the principal eigenvalue
of \eqref{eq:evpm}.
%
\begin{lemma}
\label{lem:lower-est}
For all $\lambda\geq 0$ the inequality \eqref{eq:lower-est} holds.
\end{lemma}
\begin{proof}
As before let $\mu_n(\lambda)$ denote the principal eigenvalue of
\eqref{eq:evpaux_n}. As all data are smooth we can apply
\cite[Lemma~15.6]{hess:91:ppb}, which asserts that
\begin{displaymath}
\mu_n(\lambda)\geq\mu_n(0)-\lambda\mathcal P(m_n)
\end{displaymath}
for all $\lambda\geq 0$. (Note that we used a slightly different
definition of $\mathcal P(m_n)$.) Hence, an application of
Proposition~\ref{prop:ev-conv} and Lemma~\ref{lem:limP_n} shows that
\begin{displaymath}
\mu(\lambda)\geq\mu(0)-\lambda\lim_{n\to\infty}\mathcal P(m_n)
\geq\mu(0)-\lambda\mathcal P(m),
\end{displaymath}
proving the assertion of the lemma.
\end{proof}
%
Finally, we apply the approximation procedure to get
Proposition~\ref{prop:mu}(\ref{mu-neg}).
%
\begin{lemma}
\label{lem:mu-neg}
If $\lambda\in\mathbb C$ is an eigenvalue of \eqref{eq:evpm} with
$\realpart\lambda>0$, then $\mu(\realpart\lambda)\leq 0$.
\end{lemma}
\begin{proof}
Suppose that $\lambda\in\mathbb C$ is an eigenvalue of
\eqref{eq:evpm} with $\realpart\lambda>0$. Then, by
Theorem~\ref{thm:ppm-pert}, there exists a sequence $(\lambda_n)$ of
eigenvalues to the perturbed eigenvalue problems \eqref{eq:evpm_n}
converging to $\lambda$ as $n$ tends to infinity. (The sequence
$(\lambda_n)$ is not necessarily unique.) Hence,
$\realpart\lambda_n>0$ for large $n\in\mathbb N$. As all data are
smooth, we can apply \cite[Lemma~3.6]{beltramo:84:pep} to conclude
that $\mu_n(\realpart\lambda_n)\leq 0$ for all $n\in\mathbb N$
sufficiently large. By Proposition~\ref{prop:ev-conv} $\mu_n$
converges to $\mu$ locally uniformly, and thus
$0\geq\lim_{n\to\infty}\mu_n(\realpart\lambda_n)=\mu(\realpart\lambda)$.
This concludes the proof of the lemma.
\end{proof}
\section{Upper Estimates for the Principal Eigenvalue}
\label{sec:upper-est}
In this section we provide an upper bound for the principal eigenvalue
of
\begin{equation}
\label{eq:evp1}
\begin{aligned}
\partial_t u+\mathcal A(t) u
&=\mu u&&\text{in }\Omega\times[0,T]\\
u(\cdot\,,t)
&=0&&\text{on }\partial\Omega\times [0,T]\\
u(\cdot,0) &= u(\cdot,T)&\qquad&\text{in }\Omega
\end{aligned}
\end{equation}
which leads to Proposition~\ref{prop:m-upper-est}. Throughout we
suppose that Assumption~\ref{ass:A} holds, and that $\Omega$ is a
bounded domain. Moreover, we define $A$ and $b$ as in \eqref{eq:Ab}.
Then we can rewrite $\mathcal A(t)u$ by
\begin{displaymath}
\mathcal A(t)u
=-\divergence\bigl((\nabla u)A\bigr)+(\nabla u)b+c_0 u.
\end{displaymath}
For $k\in\mathbb N\cup\{\infty\}$ we define
\begin{displaymath}
C_T^k(\bar\Omega\times\mathbb R)
:=\bigl\{u\in C^k(\bar\Omega\times\mathbb R)\colon
\text{$u(x,t+T)=u(x,t)$ for all $(x,t)\in\bar\Omega\times\mathbb R$}
\bigr\}.
\end{displaymath}
%
The following lemma is a variation of
Hess~\cite[Proposition~3.1]{hess:84:pss}. The main difference is that
in our case $\mathcal A(t)$ is in divergence form. Our aim is to give
a version for arbitrary bounded domains and operators $\mathcal A(t)$
with bounded and measurable coefficients. To achieve this we first
look at the corresponding problem in the smooth case and then pass to
the general case by the approximation procedure established in
Section~\ref{sec:approx} and the perturbation results in
Section~\ref{sec:pert}.
%
\begin{lemma}
\label{lem:basicest}
Suppose that $\Omega$ is of class $C^\infty$, and that the
coefficients of $\mathcal A(t)$ are in
$C_T^\infty(\Omega\times\mathbb R)$. Moreover, let
$\varphi\in\mathcal D(\Omega)$ be nonnegative such that
\begin{equation}
\label{eq:int1}
T\int_\Omega\varphi^2\,dx=1.
\end{equation}
Finally define $w\in C^\infty_T(\bar\Omega\times\mathbb R,\mathbb
R^N)$ by $w:=\varphi b+2A(\nabla \varphi)^\tp$. Then the principal
eigenvalue, $\mu$, of \eqref{eq:evp1} satisfies the estimate
\begin{equation}
\label{eq:muest}
\mu\leq
\frac{1}{4}\int_0^T\int_\Omega w^\tp A^{-1}w+\varphi^2 c_0\,dx\,dt.
\end{equation}
\end{lemma}
\begin{proof}
Let $u\in C_T^\infty(\bar\Omega\times\mathbb R)$ denote an
eigenfunction of \eqref{eq:evp1} to the principal eigenvalue $\mu$.
We can choose $u$ such that $u(x,t)>0$ for all
$(x,t)\in\Omega\times[0,T]$. By this choice of $u$ the function
$\psi\in C_T^\infty(\Omega\times\mathbb R)$, given by
$\psi(x,t):=-\log u(x,t)$ for all $(x,t)\in\Omega\times[0,T]$, is
well defined. As $u$ is an eigenfunction of
\eqref{eq:evp1} we get that
\begin{displaymath}
-\frac{\partial}{\partial t}\psi
=\frac{1}{u}\frac{\partial}{\partial t}u
=\mu-\frac{1}{u}\mathcal A(t)u,
\end{displaymath}
and thus by definition of $\psi$ and $\mathcal A(t)$
\begin{displaymath}
\begin{split}
\frac{1}{u}\mathcal A(t)u &=-\frac{1}{u}\divergence\bigl((\nabla
u)A\bigr)
+\frac{1}{u}(\nabla u)b+c_0\\
&=-\divergence\Bigl(\frac{1}{u}(\nabla u)A\Bigr)
-\frac{1}{u^2}(\nabla u)A(\nabla u)^\tp-(\nabla\psi) b+c_0\\
&=\divergence\bigl((\nabla \psi)A\bigr) -(\nabla
\psi)A(\nabla\psi)^\tp-(\nabla\psi) b+c_0.
\end{split}
\end{displaymath}
Combining the above two identities we see that
\begin{displaymath}
\mu=-\frac{\partial}{\partial t}\psi
+\divergence\bigl((\nabla \psi)A\bigr)
-(\nabla \psi)A(\nabla\psi)^\tp-(\nabla\psi) b+c_0.
\end{displaymath}
Next we multiply the above equation by $\varphi^2$ and integrate
over $\Omega\times(0,T)$. We can do this because $\varphi$ has
compact support in $\Omega$, and $u$ is bounded away from zero on
the support of $\varphi$. Taking into account our assumption
\eqref{eq:int1} we get that
\begin{multline*}
\mu=-\int_0^T\int_\Omega \varphi^2\frac{\partial}{\partial
t}\psi\,dx\,dt +\int_0^T\int_\Omega
\varphi^2\divergence\bigl((\nabla \psi)A\bigr)\,dx\,dt\\
-\int_0^T\int_\Omega \varphi^2(\nabla \psi)A(\nabla\psi)^\tp
+\varphi^2(\nabla\psi) b-\varphi^2c_0\,dx\,dt.
\end{multline*}
As $\psi$ is $T$-periodic in $t\in\mathbb R$ and $\varphi$ is
independent of $t$ the first integral on the right hand side of the
above identity is zero. The second integral can be rewritten as
\begin{multline*}
\int_0^T\int_\Omega
\varphi^2\divergence\bigl((\nabla \psi)A\bigr)\,dx\,dt\\
=\int_0^T\int_\Omega \divergence\bigl(\varphi^2(\nabla
\psi)A\bigr)\,dx\,dt -\int_0^T\int_\Omega 2\varphi(\nabla
\psi)A(\nabla\varphi)^\tp\,dx\,dt.
\end{multline*}
As $\varphi$ has compact support an application of the divergence
theorem shows that the first integral on the right hand side of the
above equation is zero, and thus
\begin{equation}
\label{eq:muest2}
\mu=-\int_0^T\int_\Omega(\nabla \psi)A(\nabla\psi)^\tp
+2\varphi(\nabla \psi)A(\nabla\varphi)^\tp+(\nabla\psi) b\,dx\,dt
+\int_0^T\int_\Omega c_0\,dx\,dt.
\end{equation}
We next estimate the first of the above integrals by a quantity
independent of $\psi$. To do so first note that by the ellipticity
condition \eqref{eq:Aelliptic} the matrix $A$ is invertible, and
hence $v:=\varphi(\nabla\psi)^\tp+\tfrac{1}{2}A^{-1}w$ is well
defined. Recalling that $w=\varphi b +2A(\nabla\varphi)^\tp$ and
that $A$ is symmetric, an elementary calculation shows that
\begin{equation}
\label{eq:vAv}
v^\tp Av
=\varphi^2(\nabla\psi)A(\nabla\psi)^\tp
+\tfrac{1}{4}w^\tp A^{-1}w
+\varphi^2(\nabla\psi)b+2(\nabla\varphi)A(\nabla\psi)^\tp.
\end{equation}
Clearly $v^\tp Av\geq 0$ by the ellipticity assumption
\eqref{eq:Aelliptic}. If we add $\int_0^T\int_\Omega v^\tp
Av\,dx\,dt$ to the right hand side of \eqref{eq:muest2} and take
into account \eqref{eq:vAv} we immediately arrive at
\eqref{eq:muest}, concluding the proof of the lemma.
\end{proof}
%
In the calculations in the above proof it was quite essential that
$\varphi$ does not depend on $x\in\Omega$. This can be relaxed a
little bit by looking at a transformed problem.
%
\begin{lemma}
\label{lem:upper-est}
Suppose that $\gamma\in C^1(\mathbb R,\mathbb R^N)$ is $T$-periodic.
Further assume that $\varphi_0\in\mathcal D(\Omega)$ is a
nonnegative function satisfying \eqref{eq:intpos}. Also assume that
$\varphi(x,t):=\varphi_0(x-\gamma(t))\in\Omega\times\mathbb R$ for
all $(x,t)\in\supp(\varphi_0)\times\mathbb R$, and set
\begin{displaymath}
w:=\varphi(b-d\gamma/dt)+2A(\nabla \varphi)^\tp.
\end{displaymath}
Then the principal eigenvalue, $\mu$, of \eqref{eq:evp1} satisfies
the estimate \eqref{eq:muest}.
\end{lemma}
\begin{proof}
Define the diffeomorphism $\theta\in {C^1(\mathbb R^{N+1},\mathbb
R^{N+1})}$ by $\theta(x,t):=(x-\gamma(t),t)$ for all
$(x,t)\in\mathbb R^N\times\mathbb R$. Then the inverse of $\theta$
is given by $\theta^{-1}(y,t)=(y+\gamma(t),t)$. Next set
$Q_T:=\theta(\Omega\times(0,T))$, and define
\begin{displaymath}
\mathcal A_\gamma(t)v
:=-\divergence(\nabla v (A\circ\theta^{-1})
+\nabla v(b\circ\theta^{-1}-\dot\gamma)
+(c_0\circ\theta^{-1})v.
\end{displaymath}
Suppose now that $u$ is a positive principal eigenfunction to the
principal eigenvalue $\mu$ of \eqref{eq:evp1}. Then, using that $u$
is an eigenfunction of \eqref{eq:evp1}, a simple calculation shows
that the function $v:=u\circ\theta^{-1}$ satisfies the equation
\begin{displaymath}
\frac{\partial}{\partial t}v+\mathcal A_\gamma v=\mu v
\end{displaymath}
in $Q_T$. By our assumptions we have that
$\supp(\varphi_0)\times(0,T)\subset Q_T$. Therefore we can apply
Lemma~\ref{lem:basicest} to conclude that
\begin{displaymath}
\mu\leq
\int_0^T\int_{\supp(\varphi_0)}
(w\circ\theta^{-1})^\tp (A^{-1}\circ\theta^{-1})(w\circ\theta^{-1})
+c_0\circ\theta^{-1}\varphi_0\,dy\,dt.
\end{displaymath}
(Note that in the proof of Lemma~\ref{lem:basicest} we the did not
use the boundary conditions, but only the fact that $u$ is positive
in $\Omega\times[0,T]$.) As $\det D\theta=1$ we can apply the
transformation formula for integrals and the definition of
$\varphi$ to get \eqref{eq:muest}.
\end{proof}
%
Next we get rid of the smoothness assumptions on the domain and the
coefficients of $\mathcal A(t)$. The idea is to use the approximation
procedure from Section~\ref{sec:approx}, and then the perturbation
results in Section~\ref{sec:pert}.
%
\begin{proposition}
\label{prop:upper-est}
Suppose that the assumptions of Lemma~\ref{lem:upper-est} are
satisfied, but that $\Omega\subset\mathbb R^N$ is an arbitrary
bounded domain, and that the coefficients of $\mathcal A(t)$ are
only bounded and measurable (Assumption~\ref{ass:A}). Then the
assertions of Lemma~\ref{lem:upper-est} remain true.
\end{proposition}
\begin{proof}
Suppose that $\mathcal A_n(t)$ and $\Omega_n$ are as constructed in
Section~\ref{sec:approx}. If we define $A_n$ and $w_n$ accordingly
we see from Lemma~\ref{lem:upper-est} that the principal eigenvalue,
$\mu_n$, of
\begin{displaymath}
\begin{aligned}
\partial_t u+\mathcal A_n(t) u
&=\mu u&&\text{in }\Omega_n\times[0,T]\\
u(\cdot\,,t)
&=0&&\text{on }\partial\Omega_n\times [0,T]\\
u(\cdot,0) &= u(\cdot,T)&\qquad&\text{in }\Omega_n
\end{aligned}
\end{displaymath}
satisfies the estimate
\begin{equation}
\label{eq:upper-est_n}
\mu_n\leq
\frac{1}{4}\int_0^T\int_\Omega w_n^\tp A_n^{-1}w_n
+\varphi^2c_0^{(n)}\,dx\,dt.
\end{equation}
for all $n\in\mathbb N$. As the inversion of a matrix is a smooth
operation, and the ellipticity constant of $\mathcal A_n$ is
uniformly bounded from below we have that $w_n^\tp A_n^{-1}w_n$
converges to $w^\tp A^{-1}w$ in $L_1(\Omega)$. Applying
Proposition~\ref{prop:ev-conv} the estimate \eqref{eq:muest}
follows from \eqref{eq:upper-est_n} by letting $n$ go to infinity.
This completes the proof of the proposition.
\end{proof}
\appendix
\section{Perturbations of the Initial Value Problem}
\label{app:dompert}
The purpose of this appendix is to show that the results in
\cite{daners:96:dpl} hold under our more general notion of domain
convergence given in Definition~\ref{def:domconv}. The only place we
need the explicit notion of domain convergence is in the proof of
Theorem~3.1, all subsequent results only use the assertions of that
theorem. If these assertions are true for our new notion of domain
convergence then all other results from \cite{daners:96:dpl} are
valid. We consider perturbations of the initial boundary value
problem
\begin{equation}
\label{eq:ibvp}
\begin{aligned}
\frac{\partial}{\partial t}u+\mathcal A(t)u&=f
&\qquad&\text{in }\Omega\times(0,T),\\
u&=0&&\text{on }\partial\Omega\times(0,T),\\
u(\cdot,0)&=u_0&&\text{in }\Omega.
\end{aligned}
\end{equation}
%
We next state \cite[Theorem~3.1]{daners:96:dpl}, and then provide the
necessary changes in its proof assuming the domains converge in the
more general sense given in Definition~\ref{def:domconv}.
%
\begin{theorem}
Suppose that $\Omega$ is a bounded open and stable set, and that
$\Omega_n$ is a sequence of domains with $\Omega_n\to\Omega$ in the
sense of Definition~\ref{def:domconv}. Moreover, assume that
$p>2N(N+2)^{-1}$, and that $u_{0n}\in L_2(\Omega_n)$ and $f_n\in
{L_2\bigl((0,T),L_p(\Omega_n)\bigr)}$ are such that
$u_{0n}\rightharpoonup u_0$ weakly in $L_2(\mathbb R^N)$ and
$f_n\rightharpoonup f$ weakly in $L_2\bigl((0,T),L_p(\mathbb
R^N)\bigr)$. Finally, suppose that $u_n$ is the weak solution of
\begin{equation}
\label{eq:ibvp_n}
\begin{aligned}
\frac{\partial}{\partial t}u+\mathcal A_n(t)u&=f_n
&\qquad&\text{in }\Omega\times(0,T),\\
u&=0&&\text{on }\partial\Omega_n\times(0,T),\\
u(\cdot,0)&=u_{0n}&&\text{in }\Omega_n.
\end{aligned}
\end{equation}
Then $u_n$ converges to $u$ strongly in ${L_2\bigl((0,T),L_q(\mathbb
R^N)\bigr)}$ for all $q\in [1,2N(N-2)^{-1})$, and weakly in
${L_2\bigl((0,T),W_2^1(\mathbb R^N)\bigr)}$. Moreover, $u$ is a weak
solution of \eqref{eq:ibvp}.
\end{theorem}
\begin{proof}
It follows in exactly the same way as in the proof of
\cite[Theorem~3.1]{daners:96:dpl} that $u_n$ is bounded in
${L_2\bigl((0,T), W_2^1(\mathbb R^N)\bigr)}$, and that it converges
to a function $u$ weakly in that space. Recall that we did not
assume that $\Omega_n$ stays in a common bounded set for all
$n\in\mathbb N$ (c.f.~Remark~\ref{rem:unbounded}). Hence we cannot
directly apply \cite[Lemma~2.1]{daners:96:dpl} to conclude that the
convergence of $u_n$ takes place strongly in
${L_2\bigl((0,T),L_q(\mathbb R^N)\bigr)}$ for all $q\in
[1,2N(N-2)^{-1})$. However, an obvious modification of the proof of
that lemma shows that for every bounded subset $B\subset\mathbb R^N$
the sequence $(u_n)$ converges to $u$ in
${L_2\bigl((0,T),L_q(B)\bigr)}$ for all $q\in [1,2N(N-2)^{-1})$. As
$u_n$ is bounded in ${L_2\bigl((0,T), W_2^1(\mathbb R^N)\bigr)}$ it
follows from the Sobolev embedding theorem that $u_n$ is bounded in
${L_2\bigl((0,T),L_r(\mathbb R^N)\bigr)}$ for all $q\in
[1,2N(N-2)^{-1})$. Fix now $q,r$ such that $1\leq q0$ such that
$\max\{f_n(a),f_n(b)\}\leq M_0$ for all $n\in\mathbb N$. This proves
the existence of a uniform upper bound. We now establish a uniform
lower bound. Setting $x_0:=(b-a)/2$, the convexity of $f_n$ implies
that $2f_n(x_0)\leq f_n(x_0+z)+f_n(x_0-z)$ for all $z\in\mathbb R$.
Using the upper bound already established we therefore get
\begin{displaymath}
\inf_{x\in[a,b]}f_n(x)\geq 2f_n(x_0)-\sup_{x\in[a,b]}f_n(x)
\geq 2f_n(x_0)-M_0
\end{displaymath}
for all $n\in\mathbb N$. As $f_n(x_0)$ is bounded this yields a
uniform lower bound. Hence, the family $(f_n)$ is bounded on
$[a,b]$.
\par
Next we prove the equi-continuity of the family $(f_n)$. Let
$I:=[\alpha,\beta]$ be a compact interval. Fix $\delta>0$ and let
$x,y\in I$ with $x0$. Therefore
\begin{displaymath}
-2M\delta^{-1}\leq\frac{f_n(y)-f_n(x)}{y-x}\leq 2M\delta^{-1}
\end{displaymath}
for all $n\in\mathbb N$ and all $x,y\in I$ with $x