The regularity problem for elliptic operators with boundary data in Hardy-Sobolev space $H{S}^1$

Abstract

Let $\Omega$ be a Lipschitz domain in ${\mathbb{R}}^n,n\ge 3,$ 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 $1<p<\infty$.

20 pages

Cited by in corpus (1)

• The Regularity problem for second order elliptic operators with complex-valued bounded measurable coefficients