Title: The Poisson problem in weighted Sobolev spaces on Lipschitz domains

Authors: Marius Mitrea and Michael Taylor

Issue: Volume 55 (2006), Issue 3, 1063-1090

Abstract: We study the Poisson problem $\Delta u - Vu = F$ with Dirichlet and Neumann boundary conditions on a Lipschitz domain $\Omega$ in a compact Riemannian manifold equipped with a rough metric tensor. We seek a solution $u$ in the weighted Sobolev space $W^{1,\alpha}_p(\Omega) := \{u \in L^p_{\mathrm{loc}}(\Omega) \mid \mathrm{dist}\,(\cdot,\partial\Omega)^{\alpha}[|u| + |\nabla u|] \in L^p(\Omega)\}$ when $F$ belongs to an analogous space $W^{-1,\alpha}_p(\Omega)$ in the case of the Dirichlet boundary condition, and when $F \in W^{1,-\alpha}_{p'}(\Omega)^{*}$, $1/p + 1/p'=1$, in the case of the Neumann boundary condition. We take Dirichlet boundary data in the Besov space $B^{p,p}_s(\partial\Omega)$, with $s = 1 -\alpha - 1/p$, and obtain sharp results on the range of indices $(s, 1/p)$ for which this problem is well posed, and a parallel result for the Neumann boundary condition. These results are related to the Sobolev-Besov estimates obtained in David Jerison and Carlos E. Kenig (The inhomogeneous Dirichlet problem in Lipschitz domains, J. Funct. Anal. 130 (1995), 161--219), Eugene B. Fabes, Osvaldo Mendez, and Marius Mitrea (Boundary layers on Sobolev-Besov spaces and Poisson's equation for the Laplacian in Lipschitz domains, J. Funct. Anal. 159 (1998), 323--368), and Marius Mitrea and Michael Taylor (Potential theory on Lipschitz domains in Riemannian manifolds: Sobolev-Besov space results and the Poisson problem, J. Funct. Anal. 176 (2000), 1--79; Sobolev and Besov space estimates for solutions to second order PDE on Lipschitz domains in manifolds with Dini or H\"older continuous metric tensors, Comm. Partial Differential Equations 30 (2005), 1--37). They also complement certain results in Vladimir Maz'ya and Tatyana Shaposhnikova (Higher regularity in the layer potential theory for Lipschitz domains, Indiana Univ. Math. J. 54 (2005), 99--142), whose reading inspired and motivated the current work.