Let $\Omega \subset R^n$ be an open bounded set. Consider the Dirichlet problem $$-\triangle u = 0, x \in \Omega, \quad u = \varphi, x \in \partial \Omega.$$ If $\varphi$ is a continuous function, then the problem is solvable for rather general $\Omega$. In fact, it suffices to assume that every point on the boundary $\partial \Omega$ is a endpoint of a segment, the other point of the segment lies outside $\Omega$. See the book 'Complex analysis' by Ahlfors.