Sharp regularity for general Poisson equations with borderline sources

Abstract

This article concerns optimal estimates for nonhomogeneous degenerate elliptic equation with source functions in borderline spaces of integrability. We deliver sharp Hölder continuity estimates for solutions to p-degenerate elliptic equations in rough media with sources in the weak Lebesgue space Lweaknp+ϵ. For the borderline case, f∈Lweaknp, solutions may not be bounded; nevertheless we show that solutions have bounded mean oscillation, in particular John–Nirenbergʼs exponential integrability estimates can be employed. All the results presented in this paper are optimal. Our approach is inspired by a powerful Caffarelli-type compactness method and it can be employed in a number of other situations.