Mathematical Research Letters

Volume 19 (2012)

Number 3

The regularity problem for elliptic operators with boundary data in Hardy–Sobolev space HS

Pages: 699 – 717

DOI: http://dx.doi.org/10.4310/MRL.2012.v19.n3.a14

Authors

Martin Dindoš (The University of Edinburgh and Maxwell Institute of Mathematical Sciences, Edinburgh, Scotland, United Kingdom)

Josef Kirsch (The University of Edinburgh and Maxwell Institute of Mathematical Sciences, Edinburgh, Scotland, United Kingdom)

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 indivergence form. We will establish that the solvability of theDirichlet regularity problem for boundary data in Hardy–Sobolevspace $\HS$ is equivalent to the solvability of the Dirichletregularity 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 theDirichlet problem with boundary data in $\text{BMO}$ is equivalentto the solvability for boundary data in $L^p(\partial\Omega)$ forsome $1<p<\infty$.

Keywords

second order elliptic equations, boundary value problems, Hardy-Sobolev spaces

2010 Mathematics Subject Classification

35J25, 42B37

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