Lipschitz regularity for fully nonlinear elliptic equations with $(p,q)$-growth

TL;DR

This paper establishes interior and global Lipschitz regularity for solutions of fully nonlinear elliptic equations with $(p,q)$-growth, showing solutions are Lipschitz continuous under small gap conditions.

Contribution

It proves Lipschitz regularity for solutions of nonlinear equations with $(p,q)$-growth, extending regularity results under specific gap conditions.

Findings

Solutions are Lipschitz continuous with small gap $q-p$.

Lipschitz regularity holds for H"older continuous solutions under improved bounds.

Results are analogous to regularity in double phase problems.

Abstract

We prove the interior and global Lipschitz regularity results for a solution of fully nonlinear equations with $(p,q)$-growth. We prove that for a small gap $q-p$, a solution is locally or globally Lipschitz continuous. We also prove that a given H\"older continuous solution is Lipschitz continuous under improved bounds for the gap. These gap conditions are similar to those required for the regularity of double phase problems in divergence form.