The 𝐿^{𝑝} regularity problem on Lipschitz domains

Abstract

This paper contains two results on the L p L^p regularity problem on Lipschitz domains. For second order elliptic systems and 1 > p > ∞ 1>p>\infty , we prove that the solvability of the L p L^p regularity problem is equivalent to that of the L p ′ L^{p^\prime } Dirichlet problem. For higher order elliptic equations and systems, we show that if p > 2 p>2 , the solvability of the L p L^p regularity problem is equivalent to a weak reverse Hölder condition with exponent p p .