The regularity problem for the Laplace equation in rough domains

Abstract

Let $\mathrm{\Omega }\subset {\mathbb{R}}^{n+1}$, $n\ge 2$ 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 $(D_{p^{\prime }})$, the Dirichlet problem for the Laplacian with boundary data in $L^{p^{\prime }}(\partial \mathrm{\Omega })$, and $(R_p)$ (resp., $({\tilde{R}}_p)$), the regularity problem for the Laplacian with boundary data in the Hajłasz Sobolev space $W^{1,p}(\partial \mathrm{\Omega })$ (resp., ${\tilde{W}}^{1,p}(\partial \mathrm{\Omega })$, the usual Sobolev space in terms of the tangential derivative), where $p\in (1,2+\mathit{\epsilon })$ and $1∕p+1∕p^{\prime }=1$. Our main result shows that $(D_{p^{\prime }})$ is solvable if and only if $(R_p)$ also is. Under additional geometric assumptions (two-sided local John condition or weak Poincaré inequality on the boundary), we prove that $(D_{p^{\prime }})\Rightarrow ({\tilde{R}}_p)$. In particular, we deduce that in bounded chord-arc domains (resp., two-sided chord-arc domains), there exists $p_0\in (1,2+\mathit{\epsilon })$ so that $(R_{p_0})$ (resp., $({\tilde{R}}_{p_0})$) 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 $L^p(\partial \mathrm{\Omega })$ to the inhomogeneous Sobolev space $W^{1,p}(\partial \mathrm{\Omega })$. Finally, we provide a counterexample of a chord-arc domain ${\mathrm{\Omega }}_0\subset {\mathbb{R}}^{n+1}$, $n\ge 3$, so that $({\tilde{R}}_p)$ is not solvable for any $p\in [1,\mathrm{\infty })$.