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

Mihalis Mourgoglou; Xavier Tolsa

partial differential equations

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

arXiv:2407.20385

summary

The paper proves that for a bounded chord-arc domain with Dini‑oscillating coefficients, if the regularity problem is solvable in some L^q (q>p), the boundary satisfies a weak p‑Poincaré inequality, and the domain contains large pieces of superdomains where the Neumann problem is uniformly solvable, then the Neumann problem is solvable in L^p.

Abstract

Let $I^c ◯ \subset R^{n + 1}$ be a bounded chord-arc domain, let $L = - div A \nabla$ be an elliptic operator in $I^c ◯$ 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 $L$ is solvable in $L^{q}$ for some $q > p$ in $I^c ◯$ , $\partial I^c ◯$ supports a weak $p$ -PoincarÃ© inequality, and $I^c ◯$ has very big pieces of superdomains for which the Neumann problem for $L$ is solvable uniformly in $L^{q}$ , then the Neumann problem for $L$ is solvable in $L^{p}$ in $I^c ◯$ .

Minor corrections. To appear in Archive for Rational Mechanics and Analysis (ARMA)

Topics & keywords

#elliptic equations #neumann problem #chord-arc domains #regularity problem #poincaré inequality #dini mean oscillationelliptic operatorL^p solvabilityboundary value problemsuperdomainsuniform estimates