Math 6410 • ODE

\begin{enumerate}
\item[(a)] For a smooth domain \(D\subset\R^n\), solve Poisson's equation
\[
  \nabla^2u=f\text{ in }D,\qquad \frac{\partial u}{\partial n}=g\text{ on }\partial D
\]
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
\[
  G(x,x')=\frac{1}{2\pi}\ln\lvert x-x'\rvert.
\]
\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 Neumann boundary condition on \(y=0\), and evaluate
the corresponding solution of part (a) when \(f=0\).
\end{enumerate}