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A relationship between the Dirichlet and Regularity Problems for elliptic equations

Zhongwei Shen
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Abstract

We study the relationship between the solvability of the $L^p$ Dirichlet problem $(D)_p$ and that of the $L^q$ regularity problem $(R)_q$ for second order elliptic equations with bounded measurable coefficients. 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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Zhongwei Shen. 2006-07-10. A relationship between the Dirichlet and Regularity Problems for elliptic equations. https://arxiv.org/abs/math/0607239

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