Boundary weak Harnack estimates and regularity for elliptic PDE in divergence form

TL;DR

This paper extends the weak Harnack inequality to boundary points for elliptic PDEs in divergence form, leading to new boundary gradient estimates and a generalized Hopf-Oleinik lemma, advancing boundary regularity theory.

Contribution

It provides the first boundary gradient estimate in divergence form and a more general Hopf-Oleinik lemma, expanding classical boundary regularity results.

Findings

Extended weak Harnack inequality to boundary points

Derived boundary gradient estimates for divergence form equations

Established a generalized Hopf-Oleinik lemma

Abstract

We obtain a global extension of the classical weak Harnack inequality which extends and quantifies the Hopf-Oleinik boundary-point lemma, for uniformly elliptic equations in divergence form. Among the consequences is a boundary gradient estimate, due to Krylov and well-studied for non-divergence form equations, but completely novel in the divergence framework. Another consequence is a new more general version of the Hopf-Oleinik lemma.