Very degenerate elliptic equations under almost critical Sobolev regularity

TL;DR

This paper establishes local Lipschitz continuity and higher differentiability of minimizers for degenerate elliptic functionals with coefficients and data in an almost critical Zygmund class, advancing regularity theory under weak regularity assumptions.

Contribution

It introduces new regularity results for degenerate elliptic equations assuming coefficients and data in an almost critical Zygmund class, a setting previously unexplored.

Findings

Proved local Lipschitz continuity of minimizers.

Established higher differentiability of solutions.

Extended regularity results to weak coefficient regularity.

Abstract

We prove the local Lipschitz continuity and the higher differentiability of local minimizers of integral functionals with non autonomous integrand which is degenerate convex with respect to the gradient variable. The main novelty here is that the results are obtained assuming that the coefficients have weak derivative in an almost critical Zygmund class and the datum f is assumed to belong to the same Zygmund class.