= Solution
For the corrected bounded-domain problem, the interior theorem for the <Perron method for the Dirichlet problem> gives a <classical solution> of the <Poisson equation>, and the <power barrier for an exterior cone> argument gives its continuous extension with the required <Dirichlet boundary condition>. Hence existence holds in $C^2(\Omega)\cap C^0(\overline\Omega)$.
If $u_1,u_2$ are two such <classical solutions>, then $w=u_1-u_2$ is a <harmonic function> with zero boundary trace. The <weak maximum principle for elliptic operators>, applied to $w$ and $-w$ on the bounded domain, gives $w\leq0$ and $-w\leq0$. Consequently
$$
\boxed{u_1=u_2.}
$$
\b[There is exactly one solution after adding boundedness of $\Omega$.] For the printed unbounded-domain version, the half-plane counterexample in part (i) disproves uniqueness; an additional condition at infinity or a suitable restricted solution class would instead have to be specified.
Back to article page
