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\title{MAT 280: Laplacian Eigenfunctions: Theory, Applications, and Computations\\
Lecture 3: Problems of Spectral Geometry (Summary)}
\author{Lecturer: Naoki Saito\\
        Scribe: Ashley Evans/Allen Xue}
\date{April 5, 2007}

\begin{document}
\maketitle

In this lecture, we will review the famous problems of spectral
geometry some of which are still open. The early history of spectral
geometry can be found in \cite{KAC} and \cite{PROTTER}.

\section{On the First Few Eigenvalues}

To get more details about the problems in this section, please read
\cite{FRIEDLANDER}, \cite{CHAVEL} and references therein.

\ \\
Let $\{ \lambda_n \}$ denote the Dirichlet-Laplacian eigenvalues
that satisfy the following equations:
\begin{equation}\label{DLEP}
  \left\{ \begin{array}{ll}
    -\varDelta\psi_n=\lambda_n\psi_n & \text{in $\Omega$} \\
    \psi_n=0 & \text{on $\partial\Omega$}.
\end{array} \right.
\end{equation}
Furthermore, let $\{ \nu_n \}$ denote the Neumann-Laplacian eigenvalues that satisfy the following equations:
\begin{equation}\label{NLEP}
  \left\{ \begin{array}{ll}
    -\varDelta\phi_n=\nu_n\phi_n & \text{in $\Omega$} \\
    \normalderiv{\phi_n}=0 & \text{on $\partial\Omega$}.
\end{array} \right.
\end{equation}
As we show in the next lecture, all the eigenvalues above are nonnegative and
can be ordered in the increasing order.  More precisely, we have:
$0 < \lambda_1 \leq \lambda_2 \leq \cdots$, and
$0 = \nu_0 < \nu_1 \leq \nu_2 \leq \cdots$.

\subsection{The Rayleigh Conjecture (1877)}
The Rayleigh Conjecture was proven independently by Faber \& Krahn
in 1923 and claims the following:\\

Let $\mathcal{B}_1^d$ be the unit ball in $\Rf^d$, and consider
a domain $\Omega\subset\Rf^d$ with $|\Omega|<\infty$.  Furthermore, assume that
$|\Omega|=|\mathcal{B}_1^d|$. Then
\[\lambda_1(\Omega)\geq\lambda_1(\mathcal{B}_1^d),\ \ \text{with equality if
and only if $\Omega$ is congruent to $\mathcal{B}_1^d$}.\]

\textbf{NOTE:} See Appendix for more about the unit ball and sphere in $\Rf^d$.

\subsection{The Payne-P\'oyla-Weinberger conjecture (1956)}
This conjecture was proven by Ashbaugh-Benguria in 1991. Under the
same assumption of the Rayleigh conjecture, it states the following:
\[\fr{\lambda_2(\Omega)}{\lambda_1(\Omega)}\leq\fr{\lambda_2(\mathcal{B}_1^d)}{\lambda_1(\mathcal{B}_1^d)},\ \ \text{with equality if and only if $\Omega$ is congruent to
$\mathcal{B}_1^d$}.\]
A more general statement is the following:
\[
\fr{\lambda_{m+1}(\Omega)}{\lambda_m(\Omega)}<\fr{\lambda_2(\mathcal{B}_1^d)}{\lambda_1(\mathcal{B}_1^d)}
\quad \text{for } m \in \N.
\]
The cases with $m=1,2,3$ were proven by Ashbaugh-Benguria in
1991-93. For $m\geq4$ the problem is still open.

\subsection{The Payne conjecture (1955)}
The Payne conjecture was proven by L. Friedlander in 1991 and states
as follows: Assume that $\partial\Omega$ has $C^1$ smoothness
(i.e., no corners), then
\[\nu_{k+1}\leq\lambda_k,\ \ k=1,2,\cdots\]
where $\nu_{k}$ and $\lambda_{k}$ are the $k$th Neumann \eqref{NLEP} and
Dirichlet \eqref{DLEP} Laplacian eigenvalues, respectively.

\subsection{Another Payne-P\'oyla-Weinberger conjecture}
This conjecture is still open.
Assume $\Omega\subset\Rf^2$ and $|\Omega| = |\mathcal{B}_1^2|$, then
\[
    \fr{\lambda_2(\Omega)+\lambda_3(\Omega)}{\lambda_1(\Omega)}\leq
    \fr{\lambda_2(\mathcal{B}_1^2)+\lambda_3(\mathcal{B}_1^2)}{\lambda_1(\mathcal{B}_1^2)}.
\]
This can be generalized to $\Omega\subset\Rf^d$ as follows:
\[
    \fr{\lambda_2(\Omega)+\cdots+\lambda_{d+1}(\Omega)}{\lambda_1(\Omega)}\leq
    \fr{\lambda_2(\mathcal{B}_1^d)+\cdots+\lambda_{d+1}(\mathcal{B}_1^d)}{\lambda_1(\mathcal{B}_1^d)}.
\]

\subsection{Yet another open problem}
Find the optimal upper bound for
\[\fr{\lambda_3(\Omega)}{\lambda_1(\Omega)},\ \ \text{with }
\Omega\subset\Rf^d,\ \ |\Omega|<\infty , \]

i.e., find \[\arg\sup_{\substack{\Omega\subset\Rf^d \\ |\Omega|<\infty}}
\fr{\lambda_3(\Omega)}{\lambda_1(\Omega)}\]

So far, for $d=2$, the largest $\lambda_3/\lambda_1$ found equals $35/11\approx
3.2$ where $\Omega$ is a rectangle with sides $\sqrt{8}$ and
$\sqrt{3}$. The best upper bound found so far is approximately $3.831$.

\section{Isospectral Problems}
For the review on isospectral problems, see e.g., \cite{CHAPMAN}.
\begin{defn}
Two domains $\Omega, \Omega' \subset \Rf^d$, with $|\Omega|<\infty,
|\Omega'|<\infty$, are said to be \textbf{isospectral} if
$\lambda_k(\Omega)=\lambda_k(\Omega')$ holds for any $k\in\N$ or
$\nu_k(\Omega)=\nu_k(\Omega')$ holds for any $k\in\N$.
\end{defn}

\textbf{Question:} Given $\Omega\subset\Rf^d$ and $|\Omega|<\infty$,
\[\text{Is}\ \ \{\Omega'\subset\Rf^d : |\Omega'|<\infty\text{~and
isospectral to~}\Omega\}\ \ \ \text{a compact set}?\]

For $d=2$, this is true. But for $d\geq3$, this is an open problem.

\ \\
\textbf{Examples of isospectral domains:}

In $\Rf^d$, these were given for $d\geq4$ by Urakawa (1982), and for
$d=2,3$, examples were given by Gordon-Webb-Wolpert (1991-92). In
both cases $\Omega,\Omega'$ have corners (or polygons). The problem
is open for a smooth domain.

\section{Appendix}
Some useful materials are given below.
\begin{itemize}
\item Let $\mathcal{B}_1^d$ be the unit ball in $\Rf^d$.
Then the volume of the unit ball is given by
$|\mathcal{B}_1^d|=\fr{2\pi^{d/2}}{d\Gamma(d/2)}$. And the surface
area of the unit ball is given by
$|S^{d-1}|=\fr{2\pi^{d/2}}{\Gamma(d/2)}$. For the derivations, see
\cite[Sec. 0B]{FOLLAND-PDE}.
\item The Gamma Function $\Gamma(x)$ is defined as:
\[\Gamma(x)\define\displaystyle\int_0^{\infty}s^{x-1}\e^{-s}\dd{s}, \qquad 0<x<\infty.\]
\item Some properties of the Gamma Function:
\begin{enumerate}
\item $\Gamma(1)=1$
\item $\Gamma(x+1)=x\Gamma(x)$
\item $\Gamma(n+1)=n!$, $n=0,1,2,\cdots$
\item $\Gamma(1/2)=\sqrt{\pi}$
\end{enumerate}
For the review of Gamma Function, see e.g., \cite[Appendix
3]{FOLLAND-FOURIER}
\end{itemize}


\begin{thebibliography}{10}

\bibitem{KAC}
{\sc M. Kac}: ``Can one hear the shape of a drum?'' {\em Amer.~Math.~Monthly}, vol.~73, no.~4, part 2, pp.~1-23, 1966.

\bibitem{PROTTER}
{\sc M.~H. Protter}: ``Can one hear the shape of a drum?~Revisited'' {\em SIAM Review}, vol.~29, no.~2, pp.~185-197, 1987.

\bibitem{FRIEDLANDER}
{\sc L. Friedlander, T. Hoffmann-Ostenhof (Eds.)}: ``ESI-Workshop on Geometrical Aspects of Spectral Theory,'' Matrei in East Tyrolia, Austria, July 5-12, 1999.

\bibitem{CHAVEL}
{\sc I. Chavel}: {\em Eigenvalues in Riemannian Geometry}, Academic Press, 1984.

\bibitem{CHAPMAN}
{\sc S. J. Chapman}: ``Drums that sound the same,'' {\em Amer. Math. Monthly}, vol.~102, no.~2, pp.~124-138, 1995.

\bibitem{FOLLAND-PDE}
{\sc G. B. Folland}: {\em Introduction to Partial Differential Equations}, 2nd Ed., Princeton Univ.~Press, 1995.

\bibitem{FOLLAND-FOURIER}
{\sc G.~B. Folland}: {\em Fourier Analysis and Its Applications}, Brooks/Cole Publishing Company, 1992.

\end{thebibliography}


\end{document}
