I have seen results concerning the uniqueness of solutions of Poissons equation $\Delta u = f$ with Dirichlet boundary conditions, but they always seem to ignore existence, with the excuse that it is more subtle. For what boundary regularity (i.e. piecewise continuously differentiable, smooth, etc...) can we be guaranteed that solutions exist? Naively, I would think piecewise continuously differentiable, as we can represent the solution in terms of its Green's function, which requires the normal derivative to exist (inside an integral, hence it can not exist on a set of measure zero). Is this correct; I've heard that if a boundary has a corner which is sufficiently sharp, a solution won't exist (although this might not be true).
What are the required boundary regularity conditions for a solution to the problem (with Dirichlet boundary conditions) to exist?