91Ó°ÊÓ

Prove that the solution of the Robin or third boundary value problem (5) for the Laplace equation is unique when \(\alpha>0\) is a constant.

Suppose \(q(x) \geq 0\) for \(x \in \Omega\) and consider solutions \(u \in C^{2}(\Omega) \cap C^{1}(\bar{\Omega})\) of \(\Delta u-q(x) u=0\) in \(\Omega\). Establish uniqueness theorems for (a) the Dirichlet problem, and (b) the Neumann problem.

For \(n=2\), use the method of reflections to find the Green's function for the first quadrant \(\Omega=\\{(x, y): x, y>0\\}\).