Math 6410 • ODE

Suppose
\[
  \nabla^2u-cu=f\quad\text{in }D,
\]
with
\[
  u=g\quad\text{on }\partial D,
\]
where \(D\) is bounded and \(\partial D\) is smooth.
\begin{enumerate}
\item[(a)] Assuming that a solution \(u\) exists, use an energy argument to show that
it is unique when \(c>0\).
\item[(b)] If \(c<0\) and \(f=g=0\), and \(D\) is the region \(r^2=x^2+y^2<1\), show that
\[
  u=a_0J_0(r\sqrt{-c}),\qquad a_0\text{ constant},
\]
is a solution as long as \(\sqrt{-c}\) is a zero of the Bessel function \(J_0(x)\),
which satisfies
\[
  \frac{d^2J_0}{dx^2}+\frac{1}{x}\frac{dJ_0}{dx}+J_0=0.
\]
\item[(c)] If \(c<0\) and \(f=g=0\), and \(D\) is the region \(1<x^2+y^2+z^2<4\), show
that there are non-trivial solutions if \(c=-n^2\pi^2\), \(n=1,2,3\ldots\).
\end{enumerate}