Math 6410 • ODE

Suppose that the eigenvalues \(\lambda_n\) and normalized eigenfunctions \(\phi_n(x)\)
of the Dirichlet problem for the Laplacian in a bounded domain \(D\) are known: that is,
\[
  \nabla^2\phi_n=\lambda_n\phi_n\quad\text{in }D,\qquad
  \phi_n=0\quad\text{on }\partial D
\]
for all integers \(n\ge0\)
\begin{enumerate}
\item[(a)] Using Green's Theorem show that \(\lambda_n<0\).
\item[(b)] Derive the eigenfunction expansion of the Green's function \(G(x,y)\) for the
Helmholtz equation
\[
  \nabla^2G+k^2G=\delta(x-y)\quad\text{in }D,\qquad G=0\quad\text{on }\partial D,
\]
assuming that \(k^2+\lambda_n\ne0\) for all \(n\ge0\).
\item[(c)] Now suppose that \(k^2+\lambda_n=0\) for some \(n\). Show how to construct a
generalized Green's function by solving
\[
  (\nabla^2-\lambda_n)G(x,y)=\delta(x-y)+c\phi_n(x)\phi_n(y)
\]
via an eigenfunction expansion with a suitable choice of \(c\).
\end{enumerate}