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\title{MAT 280: Laplacian Eigenfunctions: Theory, Applications, and Computations\\
Lecture 11: Laplacian Eigenvalue Problems for General Domains--III. Completeness
of a Set of Eigenfunctions and the Justification of the Separation of Variables}
\author{Lecturer: Naoki Saito\\
        Scribe: Alexander Sheynis/Allen Xue}
\date{May 3, 2007}

\begin{document}
\maketitle

\setcounter{section}{0}
\section{Completeness of a Set of Eigenfunctions}
\subsection{The Neumann Boundary Condition}
In this lecture, we begin examining a generalized look at the
Laplacian Eigenvalue Problem, particularly related to generalized
domains. Our goal here is to establish the notion of the
completeness of a set of eigenfunctions, which we then use to
justify separation of variables, a tool we so far have taken for
granted. Basic references for this lecture are \cite[Sec. 11.3,
11.5]{STRAUSS}, \cite[Chap. 11]{YOUNG} and \cite[Sec. 3.3]{FOLLAND}.
For more advanced treatments, see \cite[Sec. 6.5]{EVANS}.

\ \\
In the earlier lectures, we used:
\[\left\{\begin{array}{l}
\nu_j \text{ or } \lambda_j^{(N)} \text{ for the Neumann-Laplacian eigenvalues},\\
\psi_j \text{ or } \varphi_j^{(N)} \text{ for the Neumann-Laplacian eigenfunctions}\\
\end{array}\right.\]

For simplicity, let us use $(v_j,\psi_j)$ for Neumann BC and
$(\lambda_j,\varphi_j)$ for Dirichlet BC. And we number $\nu_j$ in
ascending order: $0=\nu_1<\nu_2\leq\nu_3\leq\cdots$, also recall
that $\psi_1(\bx)=\text{const}$.

\ \\
\begin{thm}
For the Neumann Condition, we define a ``trial function'' as any
$w(\bx) \in C^2(\Omega)$ such that $w(\bx) \not\equiv 0$. Then
similarly to the Dirichlet case, (MP), $\text{(MP)}_n$, RRA (see
Lecture 9), and the minimax principle are all valid.
\end{thm}

\underline{Note:} $w \in C^2(\Omega)$ means that there is no constraint at $\partial\Omega$.
In particular, $w$ may not satisfy the Neumann condition (nor the Dirichlet condition). As such,
the Neumann condition is also referred to as the {\bf{free}} condition and the Dirichlet condition
is referred to as the {\bf{fixed}} condition.

\begin{proof}
Define \[m=\min_{\substack{w \in C^2(\Omega)\\ w \not\equiv
0}}\left\{\frac{\|\nabla{w}\|^2}{\|w\|}\right\}.\] Let $u \in
C^2(\Omega)$ attain this minimum. Set $w=u+\varepsilon v$, $\forall
\ v \in C^2(\Omega)$. The similar procedure leads to (as a result of
variational calculus):

\begin{equation}\label{formula:1}
\int_{\Omega}(-\nabla{u} \cdot \nabla{v} + muv)\dd{\bx} = 0,\qquad \forall \ v \in C^2(\Omega)\\
\end{equation}

By Green's first identity,
\begin{equation}\label{*}
\eqref{formula:1}\implies \Int{\Omega}{}(\varDelta u + mu)v \dd{\bx}
= \Int{\partial\Omega}{}\normalderiv{u} v \dd{S}.
\end{equation}

\ \\
(i) Let's choose $v(\bx)=v_0(\bx)$, where

\[v_0(\bx)\define\left\{\begin{array}{ccc}
\text{arbitrary function} & \text{in} & C^2(\Omega),\\
0 & \text{on} & \partial\Omega
\end{array}\right.\]

Then the right hand side of \eqref{*} equals 0, which implies that $\varDelta{u}+mu=0$ inside $\Omega$.\\

(ii) Since $\varDelta{u}+mu=0$ in $\Omega$ for {\em\underline{any}} $v \in C^2(\Omega)$,

\[\eqref{*} = 0 = \int_{\partial\Omega} \normalderiv{u} v \dd{S} \implies \normalderiv{u}=0 \text{ on } \partial\Omega.\]

\end{proof}

\subsection{Completeness (in the $L^2$ sense)}
The notion of completeness for eigenfunctions is similar to that of
a complete basis, which many of us have seen in an analysis class
(or any class that significantly dealt with Fourier series), in
which a set of eigenfunctions is complete if a any arbitrary
function in the space of concern can be exactly represented as a
linear combination of the eigenfunctions. To this end, we begin with
the following theorem to illustrate this point.

\begin{thm}
Both the Dirichlet-Laplacian (DL) and the Neumann-Laplacian (NL)
eigenfunctions are {\bf{complete}} in the $L^2$ sense, i.e.,
$\forall \ f \in L^2(\Omega)$,

\begin{equation}\label{**}
\begin{array}{l}
\left\|f - \sum_{n=1}^N c_n \varphi_n \right\|^2 \rightarrow \text{ as } N \rightarrow \infty,\quad \text{where }c_n =
\fr{\inner{f}{\varphi_n}}{\inner{\varphi_n}{\varphi_n}}\\[8pt]
\left\|f - \sum_{n=1}^N d_n \psi_n \right\|^2 \rightarrow \text{ as } N \rightarrow \infty,\quad \text{where }d_n = \fr{\inner{f}{\psi_n}}{\inner{\psi_n}{\psi_n}}\\
\end{array}
\end{equation}
or $c_n=0$, $\forall n\in\N$ $\ \ \Leftrightarrow\ \ $ $f\equiv 0,\
\text{a.e.}$.
\end{thm}

\ \\
\textbf{Remark}: This is important since $\{\varphi_n\}$,
$\{\psi_n\}$ are not useful if they are not complete. In other
words, if $\exists f\in L^2(\Omega)$, such that
$\|f-\sum^{\infty}_{n=1}c_n\varphi_n\|>0$, then we get
$\{\varphi_n\}$ only spans a subspace of $L^2(\Omega)$.

\begin{proof}
The proof depends on the following two facts:

\ \\
(i) The existence of the minima of the Rayleigh Quotient, and

\ \\
(ii) $\lambda_n \uparrow \infty$, $\nu_n \uparrow \infty$, as $n
\rightarrow \infty$. (will be proved in the lecture of ``eigenvalue
asymptotics'')

\ \\
We'll prove \eqref{**} for all $f \in C_0^2(\Omega)$ for (DL) and
for all $f \in C^2(\Omega)$ for (NL), respectively. For an arbitrary
$f \in L^2(\Omega)$, see examples in \cite[Sec. 6.5]{EVANS} and
\cite[Chap. 11]{YOUNG}.

\ \\
Let's start with the (DL) case. Given an arbitrary function $f\in
C^2_0(\Omega)$, set

\[r_N(\bx) \define f(\bx) - \sum_{n=1}^N c_n \varphi_n(\bx) \in C_0^2(\Omega)\]\\

By the orthogonality, for $j=1,...,n$,

\[\inner{r_N}{\varphi_j} = \inner{f-\Sm{n=1}{N}c_n\varphi_n}{\varphi_j} = \inner{f}{\varphi_j} - c_j\inner{\varphi_j}{\varphi_j} = 0.\]

So, $r_N$ satisfies the conditions of $\text{(MP)}_{N+1}$. Thus, we have

\[\lambda_{N+1} = \min_{\substack{w \in C_0^2(\Omega),\ w\not\equiv 0\\ \inner{w}{\varphi_j} = 0,\ j=1,...,N}} \frac{\|\nabla{w}\|^2}{\|w\|^2} \leq \frac{\|\nabla{r_N}\|^2}{\|r_N\|^2}\]

Now we have
\begin{equation}\label{rN}
\begin{array}{lcl}
\|\nabla{r_N}\|^2  & = & \Int{\Omega}{} |\nabla(f -
\sum_{n=1}^N c_n
\varphi_n)|^2 \dd{\bx}\\[8pt]
& = & \Int{\Omega}{}|\nabla f|^2-2\Sum{n}c_n\nabla
f\cdot\nabla\varphi_n+\Sum{m}\Sum{n}c_mc_n\nabla\varphi_n\cdot\nabla\varphi_m\,\dd{\bx}
\end{array}
\end{equation}

By Green's first identity, we have
\[\Int{\Omega}{}\nabla
f\cdot\nabla\varphi_n=-\Int{\Omega}{}f\varDelta\varphi_n\,\dd{\bx}+\Int{\partial\Omega}{}\normalderiv{\varphi_n}f\,\dd{S}=\lambda_n\inner{f}{\varphi_n}.\]
\[\text{and similarly}\quad \Int{\Omega}{}\nabla\varphi_n\cdot\nabla\varphi_n=\delta_{mn}\lambda_n\|\varphi_n\|^2.\]

So, \eqref{rN} will give us

\[\|\nabla r_N\|^2=\Int{\Omega}{}|\nabla{f}|^2 \dd{\bx} - \Sm{n=1}{N}c_n^2
\lambda_n \|\varphi_n\|^2.\]

Since $\lambda_n>0$, we have

\[\|\nabla{r_N}\|^2 < \int_{\Omega} |\nabla{f}|^2 \dd{\bx} = \|\nabla{f}\|^2\]

And so,

\[\lambda_{N+1} \leq \frac{\|\nabla{r_N}\|^2}{\|r_N\|^2} \implies \|r_N\|^2 \leq \frac{\|\nabla{f}\|^2}{\lambda_{N+1}}\]

As we will show later $\lambda_{N+1} \uparrow \infty$ as $N
\rightarrow \infty$, we have that $\|r_N\|^2 \downarrow 0$ as $N
\rightarrow \infty$. The Neumann case can be derived similarly.

\end{proof}

\section{Justification of the Separation of Variables via Completeness}
Prior to all this, we used the separation of variables technique to
solve certain partial differential equations motivated
heuristically. With the idea of completeness, we can justify its use
if we have that $\{\varphi_n\}_{n \in \N}$ and $\{\psi_n\}_{n \in
\N}$ are a complete orthogonal basis.

\subsection{Separation of the Time Variable}
To justify the separation of the time variable, we examine the heat
equation for a more practical treatment of the justification. One
can adjust this argument to validate the separation of the time
variable in a more general context, but for our purposes it is not
necessary. The heat equation is modeled by:

\begin{equation}\label{heat}
\left\{\begin{array}{ll}
u_t = k\varDelta{u} & \text{ in } \Omega,\\
u = 0 & \text{ on } \partial\Omega,\\
u(\bx,0) = f(\bx) & \text{ in } \Omega.\\
\end{array}\right.
\end{equation}

A similar model can be formed for the (N) and (R) conditions.

\ \\
The separation of variables leads to

\[u(\bx,t) = \Sm{n=1}{\infty} A_n \e^{-\lambda_n k
t}\varphi_n(\bx).\]

But we will prove this by assuming certain differentiability on $u$
and $u \in H_0^1(\Omega)$ i.e. $\|u\|^2 + \|\nabla{u}\|^2 < \infty$,
instead of the separation of variables. For the details, see
\cite[Chap. 7]{EVANS}.

\begin{proof}
Let $u(\bx,t)$ be a solution to \eqref{heat}. For each $t$,
$u(\cdot,t) \in L^2(\Omega)$ and satisfies the Dirichlet condition.
Since $\{\varphi_n\}_{n \in \N}$ is complete in $L^2(\Omega)$, we
can expand $u(\cdot,t)$ as

\begin{equation}\label{uxt}
u(\bx,t) = \sum_{n=1}^{\infty} c_n(t) \varphi_n(\bx) \text{ (Note that $c(t)$ is unknown at this point)}\\
\end{equation}

Plugging \eqref{uxt} into \eqref{heat} (term-by-term differentiation of the series is assumed; derivatives taken in this manner fall into the ``weak derivative'' sense), we get

\[\sum_{n=1}^{\infty} c'_n(t) \varphi_n(\bx) = k\sum_{n=1}^{\infty} c_n(t) \varDelta{\varphi_n(\bx)} = \sum_{n=1}^{\infty} -k \lambda_n c_n(t) \varphi_n(\bx)\]

\begin{equation}\label{cn}
\implies c'_n(t) = -k \lambda_n c_n(t) \implies c_n(t) = A_n \e^{-k
\lambda_n t},\ \ A_n:\ \text{arbitrary const}.
\end{equation}
\end{proof}

\subsection{Separation of the Spatial Variables}
For simplicity, we will only justify separation of the spatial
variables for the 2D rectangle characterized by $\Omega = I_1 \times
I_2 \subset \Rf^2$ with $I_1$,$I_2 \subset \Rf$ as shown in
Figure~\ref{fig-1}.

\begin{figure}[h]
 \begin{center}
 \includegraphics[width=8cm]{lecture11.eps}
 \caption{\label{fig-1}$\Omega \subset \Rf^2$, with $I_1$,$I_2 \subset \Rf$.}
\end{center}
\end{figure}

With $\varDelta = \pdifh{2}{}{x} + \pdifh{2}{}{y} = \partial_{xx} + \partial_{yy}$, let $-\partial_{xx} \varphi_n^1(x) = \alpha_n \varphi_n^1(x)$ and $-\partial_{yy} \varphi_m^2(y) = \beta_n \varphi_n^2(y)$ with appropriate boundary conditions for the (D), (N), or (R) cases.

\begin{thm}
The set of products $\{\varphi_n^1(x) \varphi_m^2(y)\}_{(n,m) \in \N^2}$ is a {\bf{complete}} set of eigenfunctions for $-\varDelta$ in $\Omega$ with the given boundary conditions.
\end{thm}

\begin{proof}
We have

\begin{eqnarray*}
-\varDelta\left(\varphi_n^1(x) \varphi_m^2(y)\right) & = &
\left(-\partial_{xx}{\varphi_n^1(x)}\right) \varphi_m^2(y) +
\varphi_n^1 \left(-\partial_{yy}{\varphi_m^2(y)}\right)\\[8pt]
& = &  \left(\alpha_n + \beta_m\right) \varphi_n^1(x)
\varphi_m^2(y).
\end{eqnarray*}

Hence, $\left\{\left(\alpha_n + \beta_m, \varphi_n^1(x)
\varphi_m^2(y)\right)\right\}_{(n,m) \in \N^2}$ are eigenpairs, and
we clearly see that $\inner{\varphi_n^1 \varphi_m^2}{\varphi_{n'}^1
\varphi_{m'}^2} = \text{const}\cdot \delta_{nn'} \delta_{mm'}$.

We know that $\left\{\varphi_n^j\right\}_{n \in \N}$ is complete in
$I_j$, $j=1,2$, so the question is whether this 2D eigenvalue
problem has eigenfunctions other than the product form
$\{\varphi_n^1 \varphi_m^2\}$.

\ \\
Suppose $u$ is such an eigenfunction (non-product form), so $u$
satisfies $-\varDelta{u} = \lambda u$, $u|_{\partial\Omega} = 0$.

\ \\
(i) Suppose $\lambda \notin \left\{\alpha_n +
\beta_m\right\}_{(n,m) \in \N^2}$. Then $u \perp \varphi_n^1
\varphi_m^2$ via the Fundamental Theorem of the completeness of a
basis. Thus, we have

\[0 = \inner{u}{\varphi_n^1 \varphi_m^2} = \int_{I_2}\left(\int_{I_1} u(x,y) \varphi_n^1(x) \dd{x}\right) \varphi_m^2(y) \dd{y}\]

So, by the completeness of $\left\{\varphi_m^2(y)\right\}_{m \in
\N}$ in $I_2$, we must have $\int_{I_1} u(x,y) \varphi_n^1(x) \dd{x}
= 0$, for $y\in I_2$, a.e.. And by the completeness of
$\left\{\varphi_n^1(x)\right\}_{n \in \N}$ in $I_1$, $u(x,y) = 0$,
a.e..

\ \\
So such a $u$ is not an eigenfunction, which means that we must have
$\lambda \in \left\{\alpha_n + \beta_m\right\}_{(n,m) \in \N^2}$
i.e. $\lambda = \alpha_n + \beta_m$ for some $(n,m) \in \N^2$. It is
possible that $\alpha_n + \beta_m$ may have multipilicity greater
than 1, so let $\left\{\varphi_n^1 \varphi_m^2\right\}_{(n,m) \in
\Lambda}$ be the corresponding eigenfunctions where $\Lambda
\subsetneq \N^2$. Then set

\[\psi(x,y) = u(x,y) - \sum_{(n,m) \in \Lambda} c_{nm} \varphi_n^1(x) \varphi_m^2(y), \text{ with } c_{nm} = \frac{\inner{u}{\varphi_n^1 \varphi_m^2}}{\|\varphi_n^1 \varphi_m^2\|^2}.\]

Now $\psi(x,y) \perp \left\{\varphi_n^1 \varphi_m^2\right\}_{(n,m)
\in \Lambda}$ by construction. So, $\psi(x,y) \equiv 0 \ (a.e.)$ via
a similar argument as before. Thus,

\[u(x,y) = \sum_{(n,m) \in \Lambda} c_{nm} \varphi_n^1(x) \varphi_m^2(y)\]

Therefore, $\{\varphi^1_n\varphi^2_m\}_{(n,m)\in\N^2}$ is complete.
\end{proof}

\begin{thebibliography}{10}

\bibitem{STRAUSS}
{\sc W.~A. Strauss}, {\em Partial Differential Equations: An
Introduction}, Brooks/Cole Publishing Company, 1992.

\bibitem{YOUNG}
{\sc R. Young}, {\em An Introduction to Hilbert Space},
Cambridge Univ. Press, 1988.

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

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

\end{thebibliography}


\end{document}
