A Relationship Between the Dirichlet and Regularity Problems for Elliptic Equations

Let $\Cal{L}=\text{div}A\nabla$ be a real, symmetric second order elliptic operator with bounded measurable coefficients. Consider the elliptic equation $\Cal{L}u=0$ in a bounded Lipschitz domain $\Omega$ of $\Bbb{R}^n$. We study the relationship between the solvability of the $L^p$ Dirichlet problem $(D)_p$ with boundary data in $L^p(\partial \Omega)$ and that of the $L^q$ regularity problem $(R)_q$ with boundary data in $W^{1,q}(\partial \Omega)$, where $1<p,q<\infty$. It is known that the solvability of $(R)_p$ implies that of $(D)_{p^\prime}$. In this note we show that if $(D)_{p^\prime}$ is solvable, then either $(R)_p$ is solvable or $(R)_q$ is not solvable for any $1<q<\infty$.

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Published 1 January 2007