The regularity problem for elliptic operators with boundary data in Hardy-Sobolev space $HS^1$

TL;DR

This paper establishes an equivalence between the solvability of the Dirichlet regularity problem for elliptic operators with boundary data in Hardy-Sobolev space and in $H^{1,p}$ spaces, extending known duality results.

Contribution

It proves a new duality result linking Hardy-Sobolev boundary data solvability to $H^{1,p}$ data for elliptic operators on Lipschitz domains.

Findings

Solvability in Hardy-Sobolev space is equivalent to solvability in $H^{1,p}$ space.

Extends duality results from BMO and $L^p$ spaces to Hardy-Sobolev spaces.

Provides a new characterization of boundary data for elliptic PDEs.

Abstract

Let $\Omega$ be a Lipschitz domain in $\mathbb R^n,n\geq 3,$ and $L=\divt A\nabla$ be a second order elliptic operator in divergence form. We will establish that the solvability of the Dirichlet regularity problem for boundary data in Hardy-Sobolev space $\HS$ is equivalent to the solvability of the Dirichlet regularity problem for boundary data in $H^{1,p}$ for some $1<p<\infty$. This is a "dual result" to a theorem in \cite{DKP09}, where it has been shown that the solvability of the Dirichlet problem with boundary data in $\text{BMO}$ is equivalent to the solvability for boundary data in $L^p(\partial\Omega)$ for some $1<p<\infty$.