Green's function for second order elliptic equations in non-divergence form

TL;DR

This paper constructs Green functions for second order elliptic equations in non-divergence form with Dini continuous coefficients on $C^{1,1}$ domains, providing pointwise bounds for the Green functions and their derivatives.

Contribution

It introduces a method to construct Green functions under Dini mean oscillation conditions and establishes pointwise bounds in non-divergence elliptic equations.

Findings

Green function constructed under Dini condition

Pointwise bounds for Green functions obtained

Applicable to $C^{1,1}$ domains

Abstract

We construct the Green function for second order elliptic equations in non-divergence form when the mean oscillations of the coefficients satisfy the Dini condition and the domain has $C^{1,1}$ boundary. We also obtain pointwise bounds for the Green functions and its derivatives.