Estimates for Green's functions of elliptic equations in non-divergence form with continuous coefficients

TL;DR

This paper introduces a new method to establish the existence and derive pointwise estimates for Green's functions of non-divergence form elliptic operators with Dini mean oscillation coefficients, including a sharp comparison with constant coefficient cases.

Contribution

The paper develops a novel approach for Green's function estimates in non-divergence elliptic equations with Dini mean oscillation coefficients, advancing the theoretical understanding.

Findings

Established existence of Green's functions under Dini mean oscillation conditions

Derived sharp pointwise estimates for these Green's functions

Provided a comparison with Green's functions for constant coefficient equations

Abstract

We present a new method for the existence and pointwise estimates of a Green's function of non-divergence form elliptic operator with Dini mean oscillation coefficients. We also present a sharp comparison with the corresponding Green's function for constant coefficients equations.