Note on Green's functions of non-divergence elliptic operators with continuous coefficients

TL;DR

This paper refines existing results on Green's functions for non-divergence elliptic operators with Dini mean oscillation coefficients, demonstrating their asymptotic behavior near singularities matches that of constant coefficient cases.

Contribution

It improves prior work by establishing the asymptotic behavior of Green's functions under Dini mean oscillation conditions on coefficients.

Findings

Green's function exhibits the same asymptotic behavior as constant coefficient case near the pole.

The result extends the understanding of Green's functions for elliptic operators with less regular coefficients.

The paper confirms the asymptotic equivalence under Dini mean oscillation conditions.

Abstract

We improve a result in Kim and Lee (Ann. Appl. Math. 37(2):111--130, 2021): showing that if the coefficients of an elliptic operator in non-divergence form are of Dini mean oscillation, then its Green's function has the same asymptotic behavior near the pole $x_0$ as that of the corresponding Green's function for the elliptic equation with constant coefficients frozen at $x_0$.