The conormal and Robin boundary value problems in nonsmooth domains satisfying a measure condition

TL;DR

This paper establishes new $W^1_p$ estimates for elliptic boundary value problems with conormal and Robin conditions in nonsmooth domains that are close to half spaces, extending previous regularity results under weaker geometric assumptions.

Contribution

It introduces a new class of domains near half spaces with measure-based closeness, allowing for $W^1_p$ estimates in settings weaker than Reifenberg flatness and semi-convexity.

Findings

Established $W^1_p$ estimates for conormal problems with homogeneous boundary conditions.

Extended $W^1_p$ and weighted $W^1_p$ estimates to Robin boundary problems in these domains.

Demonstrated that the measure-based closeness condition is weaker than classical geometric conditions.

Abstract

We consider elliptic equations and systems in divergence form with the conormal or the Robin boundary conditions, with small BMO (bounded mean oscillation) or variably partially small BMO coefficients. We propose a new class of domains which are locally close to a half space (or convex domains) with respect to the Lebesgue measure in the system (or scalar, respectively) case, and obtain the $W^1_p$ estimate for the conormal problem with the homogeneous boundary condition. Such condition is weaker than the Reifenberg flatness condition, for which the closeness is measured in terms of the Hausdorff distance, and the semi-convexity condition. For the conormal problem with inhomogeneous boundary conditions, we also assume that the domain is Lipschitz. By using these results, we obtain the $W^1_p$ and weighted $W^1_p$ estimates for the Robin problem in these domains.