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\title{MAT 280: Laplacian Eigenfunctions: Theory, Applications, and Computations\\
Lecture 2: Sturm-Liouville Theorem, (Pre)History of the Laplacian
Eigenvalue Problem in $\Rf^d$}
\author{Lecturer: Naoki Saito\\
        Scribe: Adam Dobrin/Allen Xue}
\date{April 3, 2007}
\begin{document}
\maketitle

\setcounter{section}{-1}
\section{Review: Vibrations of A One Dimensional String}

In Lecture 1, for the problem of vibration of 1D string, depending
on the type of boundary condition (BC), we consider the following:
\begin{equation}
\label{1dstring} \left\{\begin{array}{ll}
u_{tt}=c^{2} u_{xx} & \text{for $x\in(0,\ell)$  and  $t > 0$}; \\
\left.\hspace{-2mm}\begin{array}{ll}
u(0,t)=u(\ell,t)=0 & \text{Dirichlet BC}\\
\text{or}\ \ u_x(0,t)=u_x(\ell,t)=0 & \text{Neumann BC}
\end{array}\right\} & \text{for $t \geq 0$}; \\
u(x,0)=f(x),\ \ u_{t}(x,0) = g(x) & \text{for $x\in[0,\ell]$}
\end{array} \right.
\end{equation}

%This system describes the vibration of a one-dimensional string. The
%terms $u(0,t) = u(\ell,t) = 0$ are the wave's homogeneous Dirichlet
%boundary conditions, whereas $f(x)$ and $g(x)$ are the initial
%position and initial velocity, respectively, of the wave.
\ \\
To solve for $u(x,t)$, we assume that the solution is
independent in time and space. That is, we can write our solution as
$u(x,t)=X(x)T(t)$, where $X(x)$ and $T(t)$ do not depend on each
other. After separating $T$ part, we had:
\begin{equation}
\label{1dstringSEP}
\left\{\begin{array}{ll}
-X''=\lambda X & \text{for $x\in(0,\ell)$  and  $\lambda\geq 0$}; \\
X(0)=X(\ell)=0 & \text{Dirichlet BC}\\
\text{or}\ \ X'(0)=X'(\ell)=0 & \text{Neumann BC}
\end{array} \right.
\end{equation}

%We find that the system $X(x)$ has eigenvalues
%\begin{equation} \lambda_n = \Big(\frac{n\pi}{\ell}\Big)^2
%\end{equation}
%with eigenfunctions
%\begin{equation}
%\phi_n(x)=\sqrt{\frac{2}{\ell}}sin\Big(\frac{n\pi x}{\ell}\Big)
%\end{equation}
% for all $n \in \N$.  Moreover, our functions $\{\phi_n(x)\}$ form an orthonormal basis for the space $L^2([0,\ell])$.
This is a 1-D version of the \textbf{Laplacian eigenvalue problem}:
given a general shape $\Omega\subset\Rf^d$
\begin{equation}
\label{LEP}
\left\{\begin{array}{ll}
-\varDelta u=\lambda u & \text{in }\Omega\subset\Rf^d\\
u=f & \text{on }\partial\Omega,\ \ \ \text{Dirichlet BC}\\
\text{or }\normalderiv{u}=g & \text{on }\partial\Omega,\ \ \
\text{Neumann BC}
\end{array}\right.
\end{equation}

%with one of three boundary conditions: Dirichlet ($u=0$ on $\partial \Omega$), Neumann \\($\frac{\partial u}{\partial \nu}=0$ on $\partial \Omega$),
%and Robin ($\frac{\partial u}{\partial \nu}+au=0$ on $\partial \Omega$ with $a \in \Rf$).
\textbf{REMARK:} In 1D problem \eqref{1dstringSEP}, we get
\[\left\{\begin{array}{ll} \lambda_n=\nu^2_n=\left(\dfrac{n\pi}{\ell}\right)^2 & \\
n=1,2,\ldots & \text{Dirichlet BC}\\
n=0,1,2,\ldots & \text{Neumann BC}
\end{array}\right.
\]
Notice that $\Omega=(0,\ell)$, with $|\Omega|=\ell$. Therefore, the
eigenvalues reflect the geometric information of $\Omega$, in this
1D case, the volume of $\Omega = $ the length of $\Omega = \ell$.

\ \\
In 1D, this line of work culminated in the work of Sturm and
Liouville (1836-37), which also accounts \emph{non-uniform} strings.

\section{Sturm and Liouville's work}
Given $\Omega=(a,b)$. Define
\begin{equation}
\begin{array}{rcl}
L_w^2[a,b]&\triangleq &\Big\{f\text{ supported on } [a,b]\
\Big|\text{
}\displaystyle\int_a^b\lvert f(x)\rvert^2w(x)\dd{x} < \infty,\\[8pt]
 & & \text{with } w(x)>0 \text{ and } w\in C[a,b]\Big\}.
\end{array}
\end{equation}
equipped with the weighted inner product for all $f,g\in L_w^2[a,b]$
as
\[\inner{f}{g}_w \define\int_a^bf(x)\overline{g(x)}w(x)\dd{x}.\]

Define $\mathcal{L}: L_w^2[a,b]\rightarrow L_w^2[a,b]$ such that for
$f \in L_w^2[a,b]$,
\begin{equation}
\label{L}
\mathcal{L}f \define (rf')'+pf
\end{equation}
where $r \in C^1[a,b]$, $r>0$ on $[a,b]$, and $p \in C[a,b]$ is
real-valued.

\ \\
We consider the Regular Sturm-Liouville Problem (RSLP):
\begin{equation}\label{RSLP}
\left\{\begin{array}{l}
\mathcal{L}f+\lambda wf=0\\
\mathcal{B}_1f=\mathcal{B}_2f=0
\end{array}\right.
\end{equation}
where $\mathcal{B}_jf \triangleq
\alpha_jf(a)+\alpha'_jf'(a)+\beta_jf(b)+\beta'_jf'(b)$, for $j=1,2$,
with constants $\alpha_j$, $\alpha'_j$, $\beta_j$, and $\beta'_j$.

\ \\
We say that the boundary conditions in \eqref{RSLP}
are \emph{self-adjoint} (relative to $\mathcal{L}$) if
\bdm
\left[ r(f'\overline{g}-f\overline{g'}) \right]_a^b = 0
\quad \text{for all $f, g$ satisfying $\mathcal{B}_j f = \mathcal{B}_j g = 0$,
$j=1,2$.}
\edm
For any $f, g$ belonging to a certain subspace of $L^2_w[a,b]$ and satisfying 
the self-adjoint boundary conditions, we can easily show that the differential 
operator defined in \eqref{L} satisfies
$\inner{\mathcal{L}f}{g}_w=\inner{f}{\mathcal{L}g}_w$. Such an $\mathcal{L}$ 
is called a \emph{self-adjoint operator}.

\ \\
Note that for the vibration of the one-dimensional string with
homogeneous Dirichlet boundary conditions, $r \equiv 1$, $p \equiv
0$, $\alpha_1=1$, $\alpha'_1=0$, $\beta_1=0$, $\beta'_1=0$,
$\alpha_2=0$, $\alpha'_2=0$, $\beta_2=1$, and $\beta'_2=0$.

\subsection{The Sturm-Liouville Theorem}
For every RSLP \eqref{RSLP}, the following hold:
\begin{enumerate}
\item All eigenvalues are real.
\item Eigenfunctions corresponding to distinct eigenvalues are orthogonal with respect to
$\inner{\cdot}{\cdot}_w$.
\item The eigenspace for any eigenvalue $\lambda$ is at most 2-dimensional, and 
these two eigenfunctions can be chosen to be orthogonal.
\item $\phi_n$, the $n^{th}$ eigenfunction corresponding to the $n^{th}$ distinct
eigenvalue, has $n-1$ zeros in $(a,b)$.
\item For any $f\in C^2[a,b]$ satisfying $\mathcal{B}_1f=\mathcal{B}_2f=0$ (but not necessarily $\mathcal{L}f+\lambda wf=0$), the series $\sum_{n=1}^{\infty}\inner{f}{\phi_n}_w\phi_n$ converges uniformly to $f$.
\item For any $f \in L_w^2[a,b]$, $\| f \|^2_{L^2_w([a,b])}=\sum_{n=1}^{\infty} | \inner{f}{\phi_n}_w |^2$ (Parseval's equality).
\end{enumerate}

\ \\
In higher dimensions, we shall not delve into generalities of
spatially varying coefficients, such as $r(x)$, $p(x)$ in RSLP
\eqref{RSLP}. We shall stick with the simple Laplacian eigenvalue
problem in $\Rf^d$, $d>1$.

\section{(Pre)History of the Laplacian Eigenvalue Problem in $\Rf^d$}

\subsection{The Lorentz Conjecture (from \cite{KAC})}

In late October of 1910, a Dutch physicist H.~A.~Lorentz delivered a
series of five ``Wolfskehl'' lectures (via a donation of Mr.~Wolfskehl, who intended
to pay the prize for a person who solved ``The Fermat Conjecture'') titled \emph{Old and New Problems of Physics}
at G\"{o}ttingen University in Germany. Referring to a Cambridge
physicist J.~H.~Jeans's work in radiation theory, Lorentz said:

%\ \\
\begin{quote}
``In an enclosure with a perfectly reflecting surface, there can
form standing electromagnetic waves analogous to tones over an organ
pipe: we shall confine our attention to very high overtones. Jeans
asks for the energy in the frequency interval $\dd{\nu}$.  To this end,
he calculates the number of overtones which lie between frequencies
$\nu$ and $\nu+\dd{\nu}$, and multiplies this number by the energy which
belongs to the frequency $\nu$, and which according to a theorem of
statistical mechanics, is the same for all frequencies.''

%\ \\
``It is here that there arises the mathematical problem to prove
that \emph{the number of sufficiently high overtones which lie between
$\nu$ and $\nu+\dd{\nu}$ is independent of the shape of the enclosure,
and is simply proportional to its volume.}
%\ \\
For many shapes for which calculations can be carried out, this
theorem has been verified in a Leiden dissertation.  There is no
doubt that it holds in general even for multiply connected regions.
Similar theorems for other vibrating structures, like membranes, air
masses, etc., should also hold.''
\end{quote}

\ \\
If we express the Lorentz conjecture in a vibrating membrane, it
becomes of the following form:
\begin{equation}
\begin{array}{lcl}
N(\lambda) & = & \sum_{\lambda_n<\lambda}1\\[7pt]
& = & \#\{\lambda_n:\text{
Dirichlet-Laplacian eigenvalues \eqref{LEP}} \,|\, \lambda_n<\lambda\}\\
& \sim & \dfrac{\lvert\Omega\rvert}{2\pi}\lambda.\\
\end{array}
\end{equation}

A mathematician D. Hilbert was attending these lectures and predicted as
follows: 
``This theorem would not be proved in my life time.'' But, in fact,
Hermann Weyl, a graduate student at that time, was also attending
these lectures. Weyl proved this conjecture four months later in
February of 1911.

\subsection{Weyl's work}
Let $\Omega\subset\Rf^d$ with $\lvert\Omega\rvert=\text{volume of
}\Omega=\int_{\Omega}\dd{\bx}<\infty$. Consider the vibration problem
\begin{equation}
\left\{\begin{array}{ll}
u_{tt}=\varDelta u & \text{in }\Omega\\
u(\bx,t)=0 & \text{on }\partial\Omega\\
u(\bx,0)=f(\bx) & \text{in }\Omega\\
u_t(\bx,0)=g(\bx) & \text{in }\Omega
\end{array}\right.
\end{equation}

Using the separation of variables $u(\bx,t)=X(\bx)T(t)$, we reach to
\[XT''=(\varDelta X)T, \ \ \ \ \dfrac{T''}{T}=\dfrac{\varDelta X}{X}=-\lambda\]
replacing $X$ by $u$, we get the
Dirichlet-Laplacian eigenvalue problem:
\begin{equation}\label{DLEP}
\left\{\begin{array}{ll}
-\varDelta u=\lambda u & \text{in }\Omega\\
u=0 & \text{on }\partial\Omega
\end{array}\right.
\end{equation}

Let $L=-\varDelta$, where
$\varDelta=\pdifh{2}{}{x_1}+\pdifh{2}{}{x_2}+\cdots
+\pdifh{2}{}{x_d}=\nabla\cdot\nabla$

\ \\
If $u\not\equiv 0$ satisfies \eqref{DLEP}, then it is called a
eigenfunction and the corresponding $\lambda$ is called the
eigenvalue. Define
$E_{\lambda}\define\{u\in\mathcal{D}(L) \,|\, Lu=\lambda u\}$ the
eigenspace corresponding to $\lambda$, with
$\dim E_{\lambda}=\text{multiplicity of } \lambda$.

\ \\
In this problem, $\{\lambda\}$ consists of countably finite number
of eigenvalues with finite multiplicity, i.e., we can order them as
\[\lambda_1\leq\lambda_2\leq\cdots\leq\lambda_k\leq\cdots\longrightarrow\infty\]

Let $L\varphi_k=\lambda_k\varphi_k$, $k=1,2,\ldots$. And let $f\in
C_0(\smean{\Omega})$, i.e., $f\in C(\smean{\Omega})$ and $f=0$ on
$\partial\Omega$, where $\smean{\Omega}=\Omega\cup\partial\Omega$.
In fact it is ok to assume $f\in L^2(\Omega)$.

%\ \\
Then $f=\sum^{\infty}_{k=1}\langle f,\varphi_k\rangle\varphi_k$.
This is called an \emph{eigenfunction expansion} of $f$, because
$\langle\varphi_k,\varphi_{\ell}\rangle=\delta_{k\ell}$ and
$\{\varphi_k\}_{k\in\N}$ is an orthonormal basis (ONB for short) of 
$L^2(\Omega)$, here $\inner{f}{g} \define \int_{\Omega}f(\bx)\smean{g(\bx)}\dd{\bx}$.

\ \\
Expanding the initial conditions into the eigenbasis, we get
\[f(\bx)=\sum^{\infty}_{k=1}\langle f,\varphi_k\rangle\varphi_k,\ \ \
g(\bx)=\sum^{\infty}_{k=1}\langle g,\varphi_k\rangle\varphi_k\]

Then, we get the solution to the vibration problem
\[u(\bx,t)=\displaystyle\sum^{\infty}_{k=1}\Big\{\langle
f,\varphi_k\rangle\cos{\sqrt{\lambda_k}t}+\dfrac{\langle
g,\varphi_k\rangle}{\sqrt{\lambda_k}}\sin{\sqrt{\lambda_k}t}\Big\}\varphi_k(\bx)\]

So, the key was the Laplacian eigenvalue problem \eqref{DLEP}.

\ \\
\textbf{Weyl's Theorem}
\begin{equation}
\lambda_k\sim\left(\frac{k}{C_d\lvert\Omega\rvert}\right)^{\frac{2}{d}}\
\ \ \text{as }k\rightarrow\infty
\end{equation}
where
$C_d\define(2\sqrt{\pi})^{-d}\Gamma\left(\frac{d}{2}\right)$.\\

Equivalently,
\begin{equation}
N(\lambda)\sim C_d\lvert\Omega\rvert\lambda^{d/2} \end{equation}
where $N(\lambda)=\#\{k\in\N \,|\, \lambda_k\leq\lambda\}$. This
equivalence is clear since $N(\lambda_k)=k$, so $k\sim
C_d|\Omega|\lambda^{d/2}_k$.

\ \\
\textbf{Weyl's Conjecture}
\begin{equation}
N(\lambda)=C_d|\Omega|_d
\lambda^{\frac{d}{2}}-\frac{C_{d-1}}{4}|\partial\Omega|_{d-1}
\lambda^{\frac{d-1}{2}}+o(\lambda^{\frac{d-1}{2}}).
\end{equation}
where $|\Omega|_d$ : volume in $\Rf^d$ and $|\partial\Omega|_{d-1}$
: volume in $\Rf^{d-1}$ or the area in $\Rf^d$.

This conjecture has not been completely solved yet and started
the field known as ``spectral geometry''.

\subsection{Can we hear the shape of a drum?}

In 1966, Mark Kac (Rockefeller Univ.) asked ``Can we hear the
shape of a drum?'' \cite{KAC}. In other words, ``how much can we know
about the shape (geometric information) of $\Omega$ from the
Laplacian eigenvalues $\{\lambda_n\}_{n=1}^{\infty}$?"

\ \\
Kac proceeded to show that for all bounded
$\Omega\subset\mathbb{R}^2$,
\begin{equation}
\displaystyle\sum_{n=1}^{\infty}\e^{-\lambda_nt}=\frac{\lvert\Omega\rvert}{4\pi
t}-\frac{\lvert\partial\Omega\rvert}{8\sqrt{\pi
t}}+o(t^{-\frac{1}{2}})\ \ \text{as }t\downarrow 0
\end{equation}
also
\begin{equation}
\begin{array}{l}
\displaystyle\sum_{n=1}^{\infty}\e^{-\lambda_nt}=\frac{\lvert\Omega\rvert}{4\pi
t}-\frac{\lvert\partial\Omega\rvert}{8\sqrt{\pi
t}}+\frac{1-r}{3}+o(1)\ \ \text{as }t\downarrow 0, \\
\ \\
\text{if $\Omega$ has $r$ holes and $\Omega$ and holes are
polygons.}
\end{array}
\end{equation}

In 1967, McKean and Singer generalized Kac's result to
$\Omega\subset\Rf^d$. For more about this work and the related work up to 1987, see \cite{PROTTER}.

\begin{thebibliography}{10}

\bibitem{FOLLAND}
{\sc G.~B. Folland}, {\em Fourier Analysis and Its Applications}, Sec.\ 3.5, 3.6, as well as other chapters, Brooks/Cole, 1992.
\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{STRAUSS}
{\sc W.~A. Strauss}, {\em Partial Differential Equations: An Introduction}, Chap.\ 10-11, John Wiley \& Sons, 1992.
\bibitem{YOSIDA}
{\sc K. Yosida}, {\em Lectures on Differential and Integral Equations}, Chap.\ 2, Wiley-Interscience, 1960.  Republished by Dover in 1991.
\bibitem{YOUNG}
{\sc N. Young}, {\em An Introduction to Hilbert Space}, Chap.\ 9, 10, 11, Cambridge Univ. Press, 1988.

\end{thebibliography}


\end{document}
