Extrapolation of solvability of the regularity and the Poisson regularity problems in rough domains

arXiv2306.06185MaRDI QIDQ6439833

Xavier Tolsa, Josep M. Gallegos, Mihalis Mourgoglou

Publication date: 9 June 2023

Abstract: Let $Omegasubsetmathbb{R}^{n+1}$, $ngeq2$, be an open set satisfying the corkscrew condition with compact and uniformly $n$-rectifiable boundary $partialOmega$, but without any connectivity assumption. We study the connection between solvability of the regularity problem for divergence form elliptic operators with boundary data in the Haj{l}asz-Sobolev space $M^{1,1}(partialOmega)$ and the weak-$mathcal{A}_{i}nfty$ property of the associated elliptic measure. In particular, we show that solvability of the regularity problem in $M^{1,p}(partialOmega)$ for $p>1$ implies solvability in $M^{1,1}(partialOmega)$ and, in the particular case of the Laplacian, solvability in $M^{1,1}(partialOmega)$ implies solvability in $M^{1,p}(partialOmega)$ for some $p>1$. Moreover, under the hypothesis that $partialOmega$ supports a $1$-weak Poincar'e inequality, we prove that the solvability of the regularity problem in the Haj{l}asz-Sobolev space $M^{1,1}(partialOmega)$ is equivalent to a stronger solvability in a Hardy-Sobolev space of tangential derivatives.

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