The regularity problem for the Laplace equation in rough domains

TL;DR

This paper investigates the relationship between the solvability of Dirichlet and regularity problems for the Laplace equation in rough domains with rectifiable boundaries, establishing equivalences and conditions under which these problems are solvable.

Contribution

It proves the equivalence between the solvability of the Dirichlet problem and the regularity problem in certain rough domains, and extends results to unbounded domains and invertibility of layer potentials.

Findings

Solvability of $(D_{p'})$ is equivalent to $(R_{p})$ in specified domains.

Existence of $p_0$ such that $(R_{p_0})$ is solvable in chord-arc domains.

Counterexample showing $( ilde R_p)$ is not solvable for any $p$ in a certain domain.

Abstract

Let $\Omega \subset \mathbb{R}^{n+1}$, $n\geq 2$, be a bounded open and connected set satisfying the corkscrew condition with uniformly $n$-rectifiable boundary. In this paper we study the connection between the solvability of $(D_{p'})$, the Dirichlet problem for the Laplacian with boundary data in $L^{p'}(\partial \Omega)$, and $(R_{p})$ (resp. $(\tilde R_{p})$), the regularity problem for the Laplacian with boundary data in the Haj\l asz Sobolev space $W^{1,p}(\partial \Omega)$ (resp. $\tilde W^{1,p}(\partial \Omega)$, the usual Sobolev space in terms of the tangential derivative), where $p \in (1,2+\varepsilon)$ and $1/p+1/p'=1$. Our main result shows that $(D_{p'})$ is solvable if and only if so is $(R_{p})$. Under additional geometric assumptions (two-sided local John condition or weak Poincar\'e inequality on the boundary), we prove that $(D_{p'}) \Rightarrow (\tilde R_{p})$. In particular, we deduce that in bounded chord-arc domains (resp. two-sided chord-arc domains) there exists $p_0 \in (1,2+\varepsilon)$ 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 $L^p(\partial \Omega)$ to the inhomogeneous Sobolev space $ W^{1,p}(\partial \Omega)$. Finally, we provide a counterexample of a chord-arc domain $\Omega_0 \subset \mathbb{R}^{n+1}$, $n \geq 3$, so that $(\tilde R_p)$ is not solvable for any $p \in [1, \infty)$.