Regularity for solutions of non-uniformly elliptic equations in non-divergence form

TL;DR

This paper establishes key estimates and inequalities for solutions to non-uniformly elliptic equations in non-divergence form, advancing understanding of their local behavior under general integrability conditions.

Contribution

It proves the Aleksandrov--Bakelman--Pucci estimate and develops local boundedness and weak Harnack inequalities without specific ellipticity structure assumptions.

Findings

Proved Aleksandrov--Bakelman--Pucci estimate for non-uniformly elliptic equations

Developed local boundedness results for solutions

Established weak Harnack inequality under integrability conditions

Abstract

We prove the Aleksandrov--Bakelman--Pucci estimate for non-uniformly elliptic equations in non-divergence form. Moreover, we investigate local behaviors of solutions of such equations by developing local boundedness and weak Harnack inequality. Here we impose an integrability assumption on ellipticity representing degeneracy or singularity, instead of specifying the particular structure of ellipticity.