Boundary estimates for positive solutions to second order elliptic equations

TL;DR

This paper establishes geometric conditions on domains that ensure boundary estimates for positive solutions to second order elliptic equations, providing sharp criteria even for harmonic functions.

Contribution

It introduces necessary and sufficient geometric conditions for boundary estimates of elliptic solutions, extending classical results to broader settings.

Findings

Conditions guarantee Hopf-Oleinik type estimates.

Conditions ensure boundary Lipschitz estimates.

Results are sharp for harmonic functions.

Abstract

Consider positive solutions to second order elliptic equations with measurable coefficients in a bounded domain, which vanish on a portion of the boundary. We give simple necessary and sufficient geometric conditions on the domain, which guarantee the Hopf-Oleinik type estimates and the boundary Lipschitz estimates for solutions. These conditions are sharp even for harmonic functions.