Boundary estimates for elliptic operators in divergence form with VMO coefficients

TL;DR

This paper develops boundary regularity and nondegeneracy estimates for elliptic systems with VMO coefficients, linking the regularity to the oscillations of coefficients and data.

Contribution

It introduces new boundary estimates for elliptic operators with VMO coefficients and extends Hopf-Oleinik type lemmas to these settings.

Findings

Established boundary regularity estimates for elliptic systems with VMO coefficients.

Derived nondegeneracy estimates of Hopf-Oleinik type for elliptic equations.

Expressed moduli of continuity in terms of $L^p$-mean oscillations.

Abstract

We establish boundary regularity estimates for elliptic systems in divergence form with VMO coefficients. Additionally, we obtain nondegeneracy estimates of the Hopf-Oleinik type lemma for elliptic equations. In both cases, the moduli of continuity are expressed in terms of the $L^p$-mean oscillations of the coefficients and data.