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Solvability of the Neumann problem for elliptic equations in chord-arc domains with very big pieces of good superdomains

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arxiv 2407.20385 v4 pith:7BO6M46U submitted 2024-07-29 math.AP

Mihalis Mourgoglou , Xavier Tolsa This is my paper · ORCID
classification math.AP
keywords omegamathcalproblemneumannsolvablechord-arcellipticpieces
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abstract

Let $\Omega \subset \mathbb{R}^{n+1}$ be a bounded chord-arc domain, let $\mathcal L=-{\rm div} A\nabla$ be an elliptic operator in $\Omega$ associated with a matrix $A$ having Dini mean oscillation coefficients, and let $1<p\leq 2$. In this paper we show that if the regularity problem for $\mathcal L$ is solvable in $L^q$ for some $q>p$ in $\Omega$, $\partial \Omega$ supports a weak $p$-Poincar\'e inequality, and $\Omega$ has very big pieces of superdomains for which the Neumann problem for $\mathcal L$ is solvable uniformly in $L^q$, then the Neumann problem for $\mathcal L$ is solvable in $L^p$ in $\Omega$.

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Cited by 3 Pith papers

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