The regularity problem for the Laplace equation in rough domains

This is the authors' abstract. We don't add key points for this paper.

Let Ω ⊂ Rⁿ⁺¹, n≥ 1, be a bounded open and connected set satisfying the corkscrew condition. Assume also that its boundary ∂ Ω 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 (Dp'), the Dirichlet problem for the Laplacian with boundary data in Lp'(∂ Ω), and (Rₚ) (resp. ( Rₚ)), the regularity problem for the Laplacian with boundary data in the Haj asz Sobolev space W1,p(∂ Ω) (resp. W1,p(∂ Ω), the usual Sobolev space in terms of the tangential derivative), where p ∈ (1,2+ε) and $1/p+1/p'=1$. In particular, we show that if (Dp') is solvable then so is (Rₚ), while in the opposite direction, solvability of ( Rₚ) implies solvability of (Dₛ), for all $s>p'$. Under additional geometric assumptions (two-sided local John condition or weak Poincare inequality on the boundary), we show that (Dp') ⇒ ( Rₚ) and (Rₚ)⇒ (Dₛ), 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+ε) so that (Rp₀) (resp. ( Rp₀)) is solvable. We also provide a counterexample of a chord-arc domain Ω₀ ⊂ Rⁿ⁺¹, n ≥ 3, so that ( Rₚ) is not solvable for any p ∈ [1, ∞).