Math 6420 • PDE

Suppose \(\Omega\) is some closed, bounded domain in two spatial dimensions.
\begin{enumerate}
\item[(a)] Under what conditions does the equation \(\nabla^2\phi=0\) in \(\Omega\),
\(\mathbf{n}\cdot\nabla\phi=f\) on \(\partial\Omega\), have a solution? Is it unique?
\item[(b)] Find an integral equation for \(\phi\) on the boundary \(\partial\Omega\). Be as
explicit as possible.
\item[(c)] Under what conditions does this integral equation have a solution?
\end{enumerate}