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\newcommand{\Rq}{ \dfrac{\| \nabla w \|^2}{\| w\|^2} }


\title{MAT 280: Laplacian Eigenfunctions: Theory, Applications, and Computations\\
Lectures 8+9: Laplacian Eigenvalue Problems for General Domains --
I. Eigenvalues as Minima of the Potential Energy}
\author{Lecturer: Naoki Saito\\
        Scribe: Milad Sabetimani/Allen Xue}
\date{April 24 \& 26, 2007}

\begin{document}
\maketitle In this lecture, we will consider Laplacian eigenvalue
problems for general domains. Since the explicit formulas only exist
for special domains (e.g., rectangles, disks, balls, etc.), what can
we say about $\{(\lambda_{n}, \varphi_{n})\}$ for a domain $\Omega$
of general shape?

\ \\
The basic references for this lecture are the texts by
Strauss~\cite[Sec.~11.1-11.2]{STRAUSS}, and Courant and
Hilbert~\cite[Sec.~VI.1]{COURANT-HILBERT}. For the details and the
survey up to the recent results, consult \cite{HENROT}.

\section{The Eigenvalues as the Minima of the Potential Energy}
Consider the following Dirichlet-Laplacian (DL) Problem, where
$\Omega$ is an open domain with general shape,
$|\Omega| < \infty$,
and $\partial\Omega$ is piecewise smoooth.
\begin{equation}\label{E:DL}
\left\{\begin{array}{ll}
-\varDelta u=\lambda u & \text{in $\Omega$,}\\
u=0 & \text{on $\partial\Omega$.}
\end{array}\right.
\end{equation}

In this lecture, we list
\[ 0<\lambda_{1} \leq \lambda_{2} \cdots
\leq \lambda_{n} \leq \cdots,
\]
where each eigenvalue is repeated according to its multiplicity.

\ \\
%\begin{notation}
%We make use of the following notation for $\bx \in \Rf^{2}$.
%$\nabla w(\bx)$ is called the gradient of $w(\bx)$ and is given by
%\[
%\nabla w = \left( \frac{\partial w}{\partial x_{1}}, \cdots \cdots,
%\frac{\partial w}{\partial x_{d}}  \right).
%\]
%Also,
%\[
%\left| \nabla w \right|^2 =
%\left( \frac{\partial w}{\partial x_{1}} \right)^2 + \cdots \cdots +
%\left( \frac{\partial w}{\partial x_{d}} \right)^2.
%\]
%\end{notation}

%\begin{def}\label{E:RC}
%The expression $\Rq$ is called the Rayleigh Quotient.
%\end{def}
%\begin{def}\label{D:PE}
%The potential energy of $w(\bx)$ in $\Omega$ is defined as follows:
%\[
%\frac{1}{2} \| \nabla w \|^2 =
%\frac{1}{2} \int_{\Omega} | \nabla w |^2 \, \dd{\bx}
%\]
%\end{def}

%One may also interpret the potential energy as the ``roughness'' of $w(\bx)$ in $\Omega$. \\

Now consider the following minimization problem (MP):
\begin{equation}\label{E:MP}\nonumber
m = \min_{ {w \in C_{0}^2(\Omega)\atop {w \not\equiv
0}}}\left\{\Rq\right\}\qquad\qquad \text{(MP)}
\end{equation}
where $C_{0}^2(\Omega)=\left\{\, w \in C^2(\Omega)\cond w=0\ \
\text{on }\partial\Omega\right\}$. The term $\Rq$ is called the {\em
Rayleigh quotient}. And $\frac{1}{2}\|\nabla
w\|^2=\frac{1}{2}\int_{\Omega}|\nabla w |^2 \, \dd{\bx}$ is the {\em
potential energy} or {\em roughness} of $w(\bx)$ in $\Omega$.

\ \\
Notice that if $u(\bx)$ is solution for (MP), then so is $a\cdot
u(\bx)$, where $a$ is an arbitrary nonzero constant.

\begin{thm}\label{T:first}
Let
\begin{align*}
\lambda_{1} &= m = \min_{ {w \in C_{0}^2(\Omega)\atop {w \not\equiv
0}}}{\left\{ \Rq \right\}}\ \ \text{and} \\[10pt]
\varphi_{1} &= \argmin_{ {w \in C_{0}^2(\Omega)\atop {w \not\equiv
0}}} {\left\{ \Rq \right\}}.
\end{align*}
then $-\varDelta \varphi_{1} = \lambda_{1} \varphi_{1}$.

\ \\
In other words, ``the first eigenvalue is the minimum of the
potential energy, and the first eigenfunction is the ground state
(state of the lowest energy).''
\end{thm}

\begin{proof}
From now on, we will call a function from $C^2_0(\Omega)$ a {\em
trial function}.

\ \\
Let $u$ be the solution of (MP) with minimum value $m\geq
0$. Then, for any trial function $w \in C_{0}^2(\Omega)$, we have
\[
m = \dfrac{ \int_{\Omega} | \nabla u |^2 \, \dd{\bx}}
{\int_{\Omega} | u |^2 \, \dd{\bx}}
\leq \dfrac{ \int_{\Omega} | \nabla w |^2 \, \dd{\bx}}
{\int_{\Omega} | w |^2 \, \dd{\bx}}.
\]
Let $v \in C_{0}^2(\Omega)$ be any other trial function such that
$w(\bx) = u(\bx) + \varepsilon v(\bx),$
where $\varepsilon$ is any real constant.  \\

Then define
\[
f(\varepsilon)\define \dfrac{ \int_{\Omega} | \nabla (u +
\varepsilon v) |^2 \, \dd{\bx}} {\int_{\Omega} |u + \varepsilon v|^2
\, \dd{\bx}},
\]
which has a minimum at $\varepsilon = 0$, i.e., $f^{'}(0) = 0.$

\ \\
Expanding $f(\varepsilon)$ in $\varepsilon$ yields
\[
f(\varepsilon) =  \dfrac{ \int_{\Omega} \left( | \nabla u |^2 + 2
\varepsilon \nabla u \cdot \nabla v + \varepsilon^2 | \nabla v |^2
\right) \, \dd{\bx}} {\int_{\Omega} \left( u^2 + 2 \varepsilon u v +
\varepsilon^2 v^2 \right) \, \dd{\bx}}.
\]

Using the quotient rule for differentiation and substituting $\varepsilon = 0,$
we obtain that

\[
0 = f^{'}(0) = \dfrac{ \left( \int_{\Omega} 2 \nabla u \cdot \nabla v \, \dd{\bx} \right)
\left( \int_{\Omega} u^2 \, \dd{\bx} \right)
- \left( \int_{\Omega} | \nabla u |^2 \, \dd{\bx} \right)
\left( \int_{\Omega} 2 u v \, \dd{\bx} \right) }
{ \left( \int_{\Omega} u^2 \, \dd{\bx} \right)^2 }.
\]

A simple algebraic manipulation produces
\[\int_{\Omega} \nabla u \cdot \nabla v \, \dd{\bx}=\left( \dfrac{ \int_{\Omega} | \nabla u |^2 \, \dd{\bx} }{
\int_{\Omega} u^2 \, \dd{\bx} } \right) \int_{\Omega} u v \,
\dd{\bx}= m \int_{\Omega} u v \, \dd{\bx}.
\]

\footnotetext[1]{See Lecture 4 for some details.}

Also, by Green's first identity (G1$^1$), we may write
\[
\int_{\Omega} \nabla u \cdot \nabla v \, \dd{\bx} + \int_{\Omega} v
\varDelta u \, \dd{\bx} = \int_{\partial\Omega} v\normalderiv{u} \,
\dd{S} = 0.
\]

The last equality follows from the fact that $v \in
C_{0}^2(\Omega)$, i.e., $\left. v \right|_{\partial\Omega} = 0$.
Therefore,
\[
\int_{\Omega} \left( \varDelta u + m u \right) v \, \dd{\bx} = 0.
\]

This is true for any $v \in C_{0}^2(\Omega)$. Therefore, we must
have $\varDelta u + m u = 0$, i.e., $m$ and $u$ are the eigenpair for the Dirichlet-Laplacian problem \eqref{E:DL}.

\ \\
We still need to show $m$ is actually the smallest eigenvalue, i.e., $m=\lambda_1$, $u=\varphi_1$. Let $-\varDelta v_{j} = \lambda_{j}
v_{j}$, where $\lambda_{j}$ is any eigenvalue of the
Dirichlet-Laplacian problem \eqref{E:DL}. Then, by definition,
\[
m \leq \dfrac{ \int_{\Omega} | \nabla v_{j} |^2 \, \dd{\bx} }{
\int_{\Omega} v_{j}^2 \, \dd{\bx} }= - \dfrac{ \int_{\Omega} v_{j}
\varDelta v_{j} \, \dd{\bx} }{ \int_{\Omega} v_{j}^2 \, \dd{\bx} }=
\dfrac{ \int_{\Omega} \lambda_{j} v_{j}^2 \, \dd{\bx} }{
\int_{\Omega} v_{j}^2 \, \dd{\bx} }= \lambda_{j}.
\]

The first equality follows from (G1). So, $m \leq \lambda_{j},\
\forall j$, and $m$ is an eigenvalue of \eqref{E:DL}. So $m =
\lambda_{1}$ and $u = \varphi_{1}$.
\end{proof}

In this proof, we apply the idea of calculus of variations.
Classical but excellent general references on calculus of variations
are \cite[Chap.~IV]{COURANT-HILBERT}, \cite[Part II]{MIKHLIN}, and
\cite[Chap.~II]{SMIRNOV}. Finally, an excellent treatment of
calculus of variations related to PDEs is \cite[Chap.~8]{EVANS}.

\ \\
\begin{ex}
Find $m = \displaystyle\min_{{w\in C_{0}^2(\Omega)\atop{w\not\equiv
0}}} \dfrac{ \int_{0}^{1} \left( w^{'} \right)^2 \, \dd{x} }{
\int_{0}^{1} w^2 \, \dd{x}},\quad w\in C^2_0(0,1).$

\ \\
\emph{Answer.} $m=\pi^2$, since the solution of this (MP) is
$w(x)=\sin\pi x=\varphi_1$ and $\lambda_1=\pi^2$.

\ \\
\textbf{Interesting to Note:} We can easily pick a function $w\in
C^2_0(0,1)$ to get an approximate value of $m$. For example, we can
choose $w(x) = ax(1-x)$, $a$ is an arbitrary constant. Let us
compare the true solution and this $w$. Here we choose the functions with unit $L^2$ norm. See Figure~\ref{Fig:compare}.
\begin{equation}
\begin{array}{lcl}
\sqrt{2}\sin\pi x & \Rightarrow & \Int{0}{1}((\sqrt{2}\sin\pi
x)')^2\,\dd{x}=\pi^2\approx 9.8696,\\[8pt]
\sqrt{30}x(1-x) & \Rightarrow &
\Int{0}{1}((\sqrt{30}x(1-x))')^2\,\dd{x}=10.
\end{array}
\end{equation}

\begin{figure}[!h]
\begin{center}
\includegraphics[width=8cm]{lecture8.eps} %[trim=l b r t]
\caption{Plots of (MP) solution $\sqrt{2}\sin\pi x$ and a trial
function $\sqrt{30}x(1-x)$.}
\label{Fig:compare}
\end{center}
\end{figure}
\end{ex}

\section{The Other Eigenvalues}
\begin{thm}[Minimum Principle for the $n$th Eigenvalue] Suppose that $\left\{(\lambda_{j},\varphi_{j})\right\}_{j=1}^{n-1}$ are already known.
Then
\begin{equation}\label{E:MPn}\nonumber \lambda_{n} = \min_{ {w \in
C_{0}^2(\Omega), \atop { w\not\equiv 0, \atop { \langle w,
\varphi_{j} \rangle = 0, \, \forall j \in \{1, \ldots, n-1 \} }}}} {
\left\{ \Rq \right\} }.\qquad\qquad \mathrm{(MP)}_n
\end{equation}
assuming that the minimum exists. Furthermore, the minimizing
function is $\varphi_{n}(\bx)$, i.e. the $n$th eigenfunction.
\end{thm}
Note that this theorem implies $\lambda_{n-1}\leq\lambda_{n}$,
$\forall n \geq 2$.

\ \\
\begin{proof}
By assumption, there exists $u(\bx)$ that is a solution to
$\text{(MP)}_n$. Let $m^{\ast}$ be the minimum value of
$\text{(MP)}_n$. So $\left. u \right|_{\partial\Omega} = 0$, and $u
\perp \varphi_{1},\ldots,\varphi_{n-1}$.

\ \\
As in the proof of the previous theorem, let $w(\bx) = u(\bx) +
\varepsilon v(\bx)$, where $w$ and $v$ satisfy the conditions for
$\text{(MP)}_n$. Then, exactly as before, we have

\begin{equation}\label{E:one}
\int_{\Omega} \left( \varDelta u + m^{\ast} u \right) v \, \dd{\bx}
= 0,
\end{equation}
for any $v \in C_{0}^2(\Omega)$ with
$v \perp \varphi_{1}, \ldots, \varphi_{n-1}$. \\

Now consider, for $j=1,\ldots,n-1$,
\begin{equation}\label{E:two}
\begin{array}{lcl}
\Int{\Omega}{} \left( \varDelta u + m^{*} u \right) \varphi_{j} \,
\dd{\bx}
 & \overset{(a)}{=} & \Int{\Omega}{} u \left( \varDelta \varphi_{j} + m^{*} \varphi_{j} \right) \,
 \dd{\bx}\\
 & \overset{(b)}{=} & (m^{\ast}-\lambda_j)\Int{\Omega}{}u\varphi_j\,
 \dd{\bx}\\
 & \overset{(c)}{=} & 0.
 \end{array}
\end{equation}

\footnotetext[1]{See Lecture 4 for some details.}

where (a) is derived by Green's second identity (G2$^1$), (b) is
from the fact that $\varDelta\varphi_j=-\lambda_j\varphi_j$, and (c)
is derived by the fact $u\perp \varphi_j$.

\ \\
Now let $h(\bx)$ be an arbitrary trial function and set
\begin{equation}\label{v_def}
v(\bx)=h(\bx) - \Sm{k=1}{n-1} c_{k} \varphi_{k}(\bx),\qquad c_{k}=\dfrac{ \inner{h}{\varphi_{k}}}
{ \inner{\varphi_{k}}{\varphi_{k}}}.
\end{equation}

Then $\inner{v}{\varphi_{j}} = 0$ for $j=1,\ldots,n-1$.

\ \\
Since $h, \varphi_{j} \in C_{0}^2(\Omega), \, \forall j \in\{1,
\ldots, n-1\}$, \eqref{E:one} is valid for $v$ defined in \eqref{v_def}.

\ \\
From \eqref{E:one} and \eqref{E:two},
\begin{equation}\nonumber
\Int{\Omega}{}\left( \varDelta u + m^{*} u \right) \left( v +
\sum_{k=1}^{n-1} c_{k} \varphi_{k} \right)\,\dd{\bx} =
\int_{\Omega} \left( \varDelta u + m^{*} u \right) h \,
\dd{\bx}=0,\qquad \forall h \in C_{0}^{2}(\Omega).
\end{equation}

This implies that $-\varDelta u = m^{*} u$. Similarly to the
previous theorem with induction, we can show that
$m^{\ast}=\lambda_n$, $u=\varphi_n$.
\end{proof}

\begin{rem}
The {\em existence} of the minima (MP) and $\text{(MP)}_n$ is a
delicate mathematical issue that we have avoided, which led to the
theory of Sobolev spaces. In fact, there are domains $D$ with rough
boundaries for which (MP) does not have any solution at all. For
further information, see \cite{ADAMS-FOURNIER}, \cite[Chap.
5]{EVANS}, \cite[Chap. 6]{FOLLAND} and \cite[Chap. 7-8]{LOSS-LOSS}.
Also \cite{MEYERS-SERRIN} is the paper that put the end to the
confusion of the two different definitions of the Sobolev spaces.
\end{rem}


\begin{thebibliography}{10}

\bibitem{COURANT-HILBERT}
{\sc R. Courant, D. Hilbert}, {\em Methods of Mathematical Physics},
Vol. I, Wiley-Interscience, 1953.

\bibitem{EVANS}
{\sc L.C. Evans}, {\em Partial Differential Equations},
AMS, 1998.

\bibitem{HENROT}
{\sc A. Henrot}: {\em Extremum Problems for Eigenvalues of Elliptic Operators,
Frontiers in Mathematics}, Birkh\"{a}user, 2006.

\bibitem{MIKHLIN}
{\sc S. G. Mikhlin}, {\em Mathematical Physics, An Advanced Course},
North-Holland Publ., 1970.

\bibitem{SMIRNOV}
{\sc V. I. Smirnov}, {\em A Course of Higher Mathematics}, Vol. IV,
Pergamon Press/Addison-Wesley, 1964.

\bibitem{STRAUSS}
{\sc W.~A. Strauss}, {\em Partial Differential Equations: An
Introduction}, John Wiley \& Sons, 1992.

\bibitem{ADAMS-FOURNIER}
{\sc R. A. Adams and J. J. F. Fournier}, {\em Sobolev Spaces, 2nd
Ed.}, Elsevier Science Ltd., Oxford, UK, 2003.

\bibitem{EVANS}
{\sc L. C. Evans}, {\em Partial Differential Equations}, AMS, 1998.

\bibitem{FOLLAND}
{\sc G. B. Folland}, {\em Introduction to Partial Differential
Equations}, Princeton Univ. Press, 1995.

\bibitem{LOSS-LOSS}
{\sc E. H. Loss and M. Loss}, {\em Analysis, 2nd Ed.}, AMS, 2001.

\bibitem{MEYERS-SERRIN}
{\sc N. G. Meyers and J. Serrin}: ``H=W'', {\em Proc. Nat. Acad.
Sci.}, vol.~51, pp.~1055-1056, 1964.

\end{thebibliography}

\end{document}
