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

Abstract

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