The poisson problem on Lipschitz domains

Abstract

The aim of this work is to describe the sharp ranges of indices, for which the Poisson problem for Laplacian with Dirichlet or Neumann boundary conditions is well-posed on the scales of Besov and Triebel-Lizorkin spaces on arbitrary Lipschitz domains. The main theorems we prove extend the work of D. Jerison and C. Kenig [JFA, 95], whose methods and results are largely restricted to the case p_ 1, and answer the open problem #3.2.21 on p. 121 in C. Kenig's book in the most complete fashion. When specialized to Hardy spaces, our results provide a solution of a (strengthened form of a) conjecture made by D.-C. Chang, S.Krantz and E. Stein regarding the regularity of the Green potentials on Hardy spaces in Lipschitz domains. The corollaries of our main results include new proofs and various extensions of: Hardy space estimates for Green potentials in convex domains due to V. Adolfsson, B.Dahlberg, S. Fromm, D. Jerison, G.Verchota and T.Wolff and the Lp - Lq estimates for the gradients of Green potentials in Lipschitz domains, due to B. Dahlberg.