The regularity problem for the Laplace equation in rough domains

Let R^n+1, n 1, be a bounded open and connected set satisfying the corkscrew condition. Assume also that its boundary is uniformly n-rectifiable and its measure theoretic boundary agrees with its topological boundary up to a set of n-dimensional Hausdorff measure zero. In this paper we study the equivalence between the solvability of (D₏'), the Dirichlet problem for the Laplacian with boundary data in L^p' (), and (R₏) (resp. (R₏) ), the regularity problem for the Laplacian with boundary data in the Haj asz Sobolev space W^1, p () (resp. W^1, p (), the usual Sobolev space in terms of the tangential derivative), where p (1, 2+) and 1/p+1/p'=1. In particular, we show that if (D₏') is solvable then so is (R₏), while in the opposite direction, solvability of (R₏) implies solvability of (Dₒ), for all s>p'. Under additional geometric assumptions (two-sided local John condition or weak Poincare inequality on the boundary), we show that (D₏') (R₏) and (R₏) (Dₒ), for all s>p'. In particular, our results show that in chord-arc domains (resp. two-sided chord-arc domains), there exists p₀ (1, 2+) so that (R₏䃐) (resp. (R₏䃐) ) is solvable. We also provide a counterexample of a chord-arc domain ₀ R^n+1, n 3, so that (Rₚ) is not solvable for any p [1, ).