Optimal regularity for degenerate elliptic equations with Hamiltonian terms

TL;DR

This paper proves optimal Hölder continuity estimates for the gradient of solutions to a class of degenerate elliptic equations with Hamiltonian terms, advancing the understanding of their regularity properties.

Contribution

It extends existing regularity results to more general degenerate elliptic equations with Hamiltonian terms, using adapted perturbative techniques.

Findings

Established optimal Hölder estimates for solutions' gradients.

Extended regularity theory to broader class of degenerate equations.

Addressed challenges posed by Hamiltonian terms in degeneracy regimes.

Abstract

We establish optimal, quantitative H\"oder estimates for the gradient of solutions to a class of degenerate elliptic equations with Hamiltonian terms. The presence of such lower-order terms introduces additional challenges, particularly in regimes where the gradient is either very small or very large. Our approach adapts perturbative techniques to capture the interplay between the degeneracy rate and the Hamiltonian's growth. Our results naturally extend the regularity theory developed by Araujo-Ricarte-Teixeira (Calc. Var. 53:605-625, 2015) and Birindelli-Demengel (Nonlinear Differ. Equ. Appl. 23:41, 2016) to a more general setting.