Dirichlet problem

Remarks

A solution, if it exists, is unique, by the maximum principle for harmonic functions. For the unit disc, the solution exists for every $\varphi$ and is the Poisson integral of $\varphi$ against the Poisson kernel. For a general bounded domain, Perron's method produces a candidate solution as the upper envelope of subharmonic functions lying below $\varphi$ on the boundary. The candidate takes the boundary value $\varphi$ at every regular boundary point. The problem is solvable for every continuous $\varphi$ if and only if every boundary point is regular. Every boundary point of a planar domain bounded by finitely many Jordan curves is regular. Example of failure: let $\Omega = \set{z \in \C : 0 < \abs{z} < 1}$, with $\varphi = 0$ on $\abs{z} = 1$ and $\varphi(0) = 1$. A solution would be bounded and harmonic near $0$, so it would extend to a harmonic function on the whole disc. This function has boundary values $0$, so it is $0$ everywhere, which contradicts $u(0) = 1$.