is solvable for every \(g \in C^0(\partial \Omega)\) if and only if all \(x \in \partial \Omega\) are regular .
Proof
If all points are regular the solution is given by the Perron solution
.
Otherwise a solution for \(g = |x - x_0|^2\) is a barrier for \(x_0 \in \partial \Omega\).
To see this, one should start with path-connected domains and proceed with path-connected components.