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# 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

#### 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