Math 6410 • ODE

Consider the second-order equation
\[
  \nabla^2u-cu=f\qquad \text{in }D\subset\R^n
\]
with \(c\) constant, \(u,f\in C^2(D)\) and
\[
  u=g\qquad \text{on }\partial D,
\]
where \(D\) is bounded and \(\partial D\) is smooth.
\begin{enumerate}
\item[(a)] Assuming a solution \(u\) exists, show that it is unique when \(c>0\). Establish
this using two different methods: (i) an integral method and (ii) a Maximum Principle.
\item[(b)] Suppose that \(n=2\), \(c<0\), \(f=g=0\), and \(D\) is the region
\(r^2=x^2+y^2<1\). Show that for \(a_0\) constant,
\[
  u=a_0J_0(r\sqrt{-c}),
\]
is a solution as long as \(\sqrt{-c}\) is a zero of the Bessel function \(J_0(x)\), which
satisfies
\[
  \frac{d^2J_0}{dx^2}+\frac1x\frac{dJ_0}{dx}+J_0=0.
\]
\item[(c)] Suppose that \(n=3\), \(c<0\), \(f=g=0\), 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}