Criterion for the Sobolev well-posedness of the Dirichlet problem for the Poisson equation in Lipschitz domains. II

We study the Dirichlet problem for the Poisson equation in bounded Lipschitz domains. We show that its well-posedness in the higher order Sobolev space implies a discrete Hardy type inequality that contains a positive harmonic function with vanishing trace and the approximative numbers of the boundary of the domain. This necessary condition is also expected to be sufficient for the well-posedness. A simpler condition occurring in the author's straightenability theory of Lipschitz domains is shown to be equivalent to the existence of a homeomorphism that straightens the boundary and preserves with respect to composition the subspace of zero trace functions in the considered Sobolev space.