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\newcommand{\nodal}{\mathcal{N}}
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\title{MAT 280: Laplacian Eigenfunctions: Theory, Applications, and Computations\\
Lecture 7: Nodal Sets}
\author{Lecturer: Naoki Saito\\
        Scribe: Bradley Marchand/Allen Xue}
\date{April 20, 2007}

\begin{document}

\maketitle

\setcounter{section}{1}

Consider the eigenfunctions $v(\bx)$ of the Laplacian,
\begin{equation} \label{eqn:laplacian}
    -\varDelta v=\lambda v \ \textrm{ in $\Omega$},
\end{equation}
with Dirichlet, Neumann, or Robin boundary conditions. We define the nodal set, $\nodal$, as follows:

\begin{defn}
 The nodal set, $\nodal$, is the set of points in $\Omega$ such that the eigenfunctions of \eqref{eqn:laplacian} are zero. Thus,
 \begin{displaymath}
  \nodal \define \{\bx\in\Omega \cond v(\bx)=0\}
 \end{displaymath}
\end{defn}

Note that since $\Omega$ is open, no point on the boundary of
$\Omega$ is in $\nodal$ (i.e., $\partial\Omega \not\subset \nodal$).

\ \\
Nodal sets are important because they allow us to visualize the sets
where $v(\bx)>0$ and $v(\bx)<0$. They mark a natural division of the
domain into regions. It is this property that interested a German
physicist named Ernst Chladni~\cite{CHLADNI}. Chladni's most well
known work was developing techniques for demonstrating the various
modes of vibration in a surface. By drawing a bow over a piece of
metal lightly covered in sand, Chladni was able to visually show the
nodal sets for varying frequencies. Similar techniques are
particularly important in making musical instruments, since the
symmetries in the nodal lines can be interpreted as a measure of
``quality.''

\ \\
To gain a better understanding of nodal sets let us consider a few examples.

\begin{ex}
\emph{1D Vibrating String} \\

Earlier we showed that the eigenfunctions for the 1D
Dirichlet-Laplacian eigenvalue problem in the domain of
$\Omega=(0,\ell)$ are
\begin{equation}
 v(x) = \sqrt{\frac{2}{\ell}} \sin \left( \frac{n\pi}{\ell} x
 \right)=\varphi_n(x),\quad n\in\N.
\end{equation}
From the eigenfunctions we can see that
\begin{displaymath}
 \nodal_n = \text{ Nodal set for $\varphi_n$ }=\left\{x=\frac{k\ell}{n} \ \Big| \ k=1,2,\ldots,n-1 \right\}.
\end{displaymath}
Note that $k=0$ and $k=n$ are not in $\nodal$ since they are on the boundary.

\begin{figure}
 \centering
 \includegraphics[width=3.5in]{nodal_example.eps}
 %\include{nodal_example}
 \caption{First four eigenfunctions for 1D Dirichlet-Laplacian eigenvalue problem in the domain of $\Omega=(0,\ell)$. Red points are Nodal nodes.}
 \label{fig:nodal_ex}
\end{figure}
\end{ex}

\ \\
The nodes in the nodal set can also be given a physical
interpretation. Let us consider a wave equation with the Dirichlet
boundary condition. We already know that
\begin{equation}
 u(\bx,t) = (A \cos \sqrt{\lambda} ct + B \sin \sqrt{\lambda} ct)v(\bx)
\end{equation}
is a solution for the wave equation (no initial condition is fixed
at this point). Here $v(\bx)$ is an eigenfunction of the
Dirichlet-Laplacian.

\ \\
Then for all $\bx\in\nodal$, $u(\bx,t) \equiv 0$ for all $t\geq
0$. Thus, all the nodes in the nodal set are stationary points for
all time. This has physical implications if we consider a guitar
string. If the guitar player holds his finger on the middle of the
string then the only tones that have nodes at $x=\frac{\ell}{2}$
survive (see Figure~\ref{fig:guitar_string}), which means that the
frequencies $\frac{n \pi c}{\ell}$ with even $n$ survive and that
the frequencies with odd $n$ cannot happen.

\begin{figure}
 \centering
 \includegraphics[width=3.5in]{guitar.eps}
 %\include{guitar}
 \caption{If a guitar string is held in the middle, only eigenfunctions with nodal points in the middle are possible. Thereby, limiting the possible sounds that the string can produce.}
 \label{fig:guitar_string}
\end{figure}

\begin{ex} \emph{The Square in 2D}

\ \\
Let us consider the eigenfunctions of Dirichlet-Laplacian on a 2D
square, i.e., $\Omega = \{(x,y)\cond 0<x<\pi,\ 0<y<\pi\}$.
Previously, we found that the eigenfunctions and eigenvalues are
\begin{equation}
\left\{\begin{array}{rcl}
             \varphi(x,y) & = & A \sin nx \sin my \\
             \lambda_{nm} & = & n^2+m^2 \\
        \end{array}\right.\quad \text{with $n,m\in\N$}.
\end{equation}
Since we know what the eigenvalues and functions are, we can
tabulate them in order of increasing eigenvalues. For eigenvalues
with multiple eigenfunctions (i.e. eigenvalues with multiplicity) we
consider a linear combination of the eigenfunctions.

\ \\
For example, there are two corresponding eigenfunctions
$\varphi_{12}$ and $\varphi_{21}$ for $\lambda=5$. Thus, we write
$\varphi(x,y) = A\varphi_{12}(x,y)+B\varphi_{21}(x,y)$ as the
eigenfunction for $\lambda=5$. Table~\ref{tbl:eigenvalues} contains
a few of the tabulated eigenvalues and functions. Note that by
ordering the eigenvalues in this fashion we are able to index the
eigenvalues and eigenfunctions with one index instead of two. See
the Table~\ref{tbl:eigenvalues}.

\ \\
There are some interesting questions we would like to consider.
First, how many ways can a given integer $\lambda$ be written as the
sum of two squares of integers? Secondly, we want to ask ``what do
the nodal sets corresponding to different eigenvalues look like?''

\ \\
The first question is answered by Number Theory. Specifically,
we know the following:
\begin{thm}
A positive integer $n$ can be written as the sum of two squares if
and only if no prime congruent to 3 modulo 4 appears an odd number
of times in the factorization of $n$ into primes.
\end{thm}
See \cite[Chap.\ 7.2]{EYNDEN} for details.

\begin{figure}
 \centering
 \begin{tabular}{c|l}
  $\lambda_n$ & $\varphi_n(x,y)$ \\
  \hline
  2  & $A\sin x \sin y$ \\
  5  & $A\sin 2x \sin y + B\sin x \sin 2y$ \\
  8  & $A\sin 2x \sin 2y$ \\
  10 & $A\sin 3x \sin y + B \sin x \sin 3y$
 \end{tabular}
 \caption{Table of eigenvalues and eigenfunctions for Dirichlet-Laplacian problem on the 2D square whose side length is $\pi$, where $A$, $B$ are appropriate coefficients.}
 \label{tbl:eigenvalues}
\end{figure}

\ \\
Let us investigate a bit about the second question. First we
consider eigenvalues of single multiplicity. Since the eigenvalue is
of single multiplicity we know that $\varphi(x,y)=A\sin nx \sin my$
for some $n,m$ in $\N$. Thus, we can easily see that the nodal set
will be the set $\{x: x=\pi/p\} \cup \{y: y=\pi/q\}$ where $p,q \in
\N$ such that $1<p<n$ divides $n$ and $1<q<m$ divides $m$. Those
interested in the number of divisors of $n$ and $m$ can once again
look to Number Theory for the answer \cite[Chap.\ 3]{EYNDEN}. In the
case where $\lambda=2$, we get the lines $x=\pi/2$ and $y=\pi/2$.

\begin{figure}
 \centering
 \begin{tabular}{ccc}
  \includegraphics[width=3cm]{sq_nodal1.eps} &
  \includegraphics[width=3cm]{sq_nodal2.eps} &
  \includegraphics[width=3cm]{sq_nodal3.eps} \\
  $\varphi_{31}$ & $\varphi_{31} + \varphi_{13}$ & $\varphi_{31} - \varphi_{13}$ \\
  \includegraphics[width=3cm]{sq_nodal4.eps} &
  \includegraphics[width=3cm]{sq_nodal5.eps} &
  \includegraphics[width=3cm]{sq_nodal6.eps} \\
  $\varphi_{13}$ & $\varphi_{31}+1/3\varphi_{13}$ & $\varphi_{31}+1/3 \varphi_{13}$
 \end{tabular}
 \caption{Some nodal sets for $\lambda=10$ of the Dirichlet-Laplacian on the square.}
 \label{fig:nodal_sets}
\end{figure}

\ \\
For eigenvalues of double multiplicity the nodal sets can be much
more exotic since we have a linear combination of two
eigenfunctions. When $A$ or $B$ is zero, we have the solutions that
were discribed above. For different values of $A$ and $B$ we get
different looking nodal sets. Images of nodal sets for $\lambda=10$
can be seen in Figure~\ref{fig:nodal_sets}. More images of nodal
sets on the square can be seen in \cite[Sec.\ 10.4]{STRAUSS}.
Trefethen \cite[Chap.\ 11]{TREFETHEN} provides insight into the
programming involved in solving certain differential equations
numerically.
\end{ex}
\ \\

\begin{ex} \emph{The 3D Ball}

\ \\
Recall that the eigenfunctions for the Dirichlet-Laplacian on a 3D
ball, i.e., $\Omega=\{(x,y,z)\cond x^2+y^2+z^2<a^2\}$ are
\begin{equation}
\varphi_{\ell mj}(r,\theta,\phi) =
\frac{\bessel_{\ell+1/2}(\sqrt{\lambda_{\ell
j}}r)}{\sqrt{r}}\aLegendre_{\ell}^m(\cos \theta)(A \cos m\phi +
B\sin m\phi).
\end{equation}
where we use the spherical coordinates, which were introduced in
Lecture 6.

\ \\
The nodal set for $\varphi_{\ell mj}$ is a union of the following
kinds of surfaces:
\begin{enumerate}[label=(\roman{*})]
\item Spheres inside $\Omega$, (corresponding to the zeros of the Bessel
functions);
\item Vertical planes, i.e., $\phi=\text{const}$; and
\item Horizontal planes, i.e., $\theta=\text{const}$,
\end{enumerate}
which in fact contains $j-1$ spheres, $m$ vertical planes and
$\ell-m$ horizontal planes.
\end{ex}
%These surfaces are depicted in Figure~\ref{fig:sphere}.

%\begin{figure}
% \centering
% \includegraphics[width=2.5in]{sphere.eps}
% %\include{sphere}
% \caption{The possible nodal sets of a 3D ball. Nodal sets of types (\emph{i}), (\emph{ii}), and (\emph{iii}) can be seen in green, blue and red respectively.}
% \label{fig:sphere}
%\end{figure}
%\end{ex}
\ \\
So, how many regions can the nodal set (of the
Dirichlet-Laplacian) divide a general domain $\Omega$ into (assuming
$\Omega$ is connected)? The following theorem limits the
possibilities.

\begin{thm}[Courant Nodal Domain Theorem] \

\begin{enumerate}[label=(\roman{*})]
\item The first eigenfunction, $\varphi_{1}(x)$ corresponding to the smallest eigenvalue, $\lambda_1$, cannot have any
nodes.
\item For $n\geq 2$, $\varphi_{n}(x)$ corresponding to the $n$th eigenvalue counting multiplicity, divides the domain $\Omega$ into at least 2 and at most $n$
pieces.
\end{enumerate}
\end{thm}

Discussions on nodal sets and the ``Courant Nodal Domain Theorem''
can be found in \cite[Vol.\ I, Sec.\ V.5, VI.6]{COURANT-HILBERT} and
\cite[Sec.\ 10.4]{STRAUSS}. We will prove this later in this course
when we deal with the general eigenvalue problem using the calculus
of variations.

\ \\
At this point it is easy to show that $\varphi_n(\bx)$ must divide
$\Omega$ into at least 2 pieces if $n \geq 2$ if we assume
(\emph{i}) is correct.

\begin{proof}
We know that $\varphi_1$ is perpendicular to $\varphi_n$ for $n \geq
2$. So,
\begin{displaymath}
\Int{\Omega}{} \varphi_1(\bx)\varphi_n(\bx) \dd{\bx}=0
\end{displaymath}
and from (\emph{i}) we know that $\varphi_1(\bx)>0$ or
$\varphi_1(\bx)<0$ in $\Omega$. Thus, $\varphi_n(\bx)$ must change
its sign in $\Omega$. So there exist zeros of $\varphi_n(\bx)$ in
$\Omega$ by the continuity of $\varphi_n$. These zeros form a nodal
set.
\end{proof}

\textbf{REMARK:} \\

In 1950, Szeg\H{o} conjectured a similar theorem to the Courant
Nodal Domain Theorem for the biharmonic eigenvalue problem,
\begin{equation}\label{eqn:biharmonic}
 \left\{\begin{array}{ll}
             \varDelta^2 u = \lambda u & \textrm{in } \Omega, \\
             u = \normalderiv{u}= 0 & \textrm{on } \partial
             \Omega.
        \end{array}\right.
\end{equation}
Szeg\H{o}'s conjecture was the following:
\begin{conj}[Szeg\H{o}, 1950]
 If $\Omega \subset \Rf^2$ is a ``nice'' domain (i.e., $\partial\Omega$ is an analytic curve), then $\varphi_1$ for \eqref{eqn:biharmonic} does not change its sign.
\end{conj}
However, surprisingly, the conjecture is not even true for the first
eigenfunction. For the details and its history including Szeg\H{o}'s
conjecture see \cite{SWEERS}.

\begin{thebibliography}{1}

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

\bibitem{EYNDEN}
{\sc C.~V. Eynden}, {\em Elementary Number Theory}, McGraw-Hill,
2001.

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

\bibitem{SWEERS}
{\sc G. Sweers}, ``When is the first eigenfunction for the clamped
plate equation of fixed sign?'', {\em Election. J. Diff. Eqns.},
Conf. 06, 2001, pp.~285-296.

\bibitem{TREFETHEN}
{\sc L.~N. Trefethen}, {\em Spectral Methods in MATLAB}, SIAM, 2000.

\bibitem{CHLADNI}
To know about Ernst Chladni, search on http://en.wikipedia.org/
\end{thebibliography}


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