Math 6410 • ODE

\begin{enumerate}
\item[(a)] Solve Poisson's equation
\[
  \nabla^2u=f\text{ in }D,\qquad u=g\text{ on }\partial D
\]
in terms of an appropriately defined Green's function.

\item[(b)] Show that the two--dimensional Green's function for the Laplacian in an
unbounded domain is
\[
  G(\mathbf{x},\mathbf{x}')=\frac{1}{2\pi}\ln\lvert\mathbf{x}-\mathbf{x}'\rvert.
\]

\item[(c)] Use the method of images to derive the Green's function for the Laplacian
in the half--space \(x\in\R\), \(y>0\) with a Dirichlet boundary condition on \(y=0\),
and evaluate the corresponding solution of part (a) when \(f=0\).
\end{enumerate}