Mihalis Mourgoglou, Xavier Tolsa

Duke Math. J. 173 (9), 1731-1837, (15 June 2024) DOI: 10.1215/00127094-2023-0044
KEYWORDS: Laplace equation, regularity problem, Dirichlet problem, chord-arc domain, 31B15, 35J25, 42B25, 42B37

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 $1\u2215p+1\u2215{p}^{\prime}=1$. 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 $p\in [1,\mathrm{\infty})$.