The Regularity Problem For The Laplace Equation In Rough Domains

Abstract

Let Omega subset of Rn+1, n >= 2 be a bounded open and connected set satisfying the corkscrew condition with uniformly n-rectifiable boundary. In this paper we study the connection among the solvability of (D-p '), the Dirichlet problem for the Laplacian with boundary data in L-p '(partial derivative Omega), and (R-p) (resp., (R-p)), the regularity problem for the Laplacian with boundary data in the Haj & lstrok;asz Sobolev space W-1,W-p (partial derivative Omega) (resp., W-1,W-p(partial derivative Omega), the usual Sobolev space in terms of the tangential derivative), where p is an element of (1, 2 + epsilon) and 1/p + 1/p ' = 1. Our main result shows that (D-p ') is solvable if and only if (R-p) also is. Under additional geometric assumptions (two-sided local John condition or weak Poincar & eacute; inequality on the boundary), we prove that (D-p ') double right arrow (R-p). In particular, we deduce that in bounded chord-arc domains (resp., two-sided chord-arc domains), there exists p(0 )is an element of (1, 2 + epsilon) so that (R-p0) (resp., (R-p0)) is solvable. We also extend the results to unbounded domains with compact boundary and show that in two-sided corkscrew domains with n-Ahlfors-David regular boundaries, the single-layer potential operator is invertible from L-p(partial derivative Omega) to the inhomogeneous Sobolev space W-1,W-p(partial derivative Omega). Finally, we provide a counterexample of a chord-arc domain Omega(0 )subset of Rn+1, n >= 3, so that (R-p) is not solvable for any p is an element of [1, infinity).