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

We study the Dirichlet problem for the Poisson equation in bounded Lipschitz domains. We show that its well-posedness in the first order Sobolev space is equivalent to the condition of K. Nyström (1996). This criterion is simpler than the similar criterion of Z. Shen (2005) due to using one positive harmonic function with vanishing trace instead of gradients of all harmonic functions with vanishing trace. Our criterion yields the main known facts about this well-posedness except for Shen's criterion. Finally, we determine all possible combinations of three basic properties (injectivity, denseness of range and closedness of range) of the operator of the boundary value problem under consideration.