Lipschitz regularity results for nonlinear strictly elliptic equations and applications

TL;DR

This paper establishes new Lipschitz regularity results for a broad class of nonlinear strictly elliptic equations with arbitrary growth in the gradient, using Ishii-Lions' method, and applies these results to ergodic problems and evolution equations.

Contribution

It introduces novel Lipschitz regularity results for equations with arbitrary gradient growth, expanding beyond previous subquadratic constraints, via Ishii-Lions' approach.

Findings

Lipschitz bounds for equations with superquadratic growth

Application to ergodic problem solutions

Analysis of large time behavior of evolution equations

Abstract

Most of lipschitz regularity results for nonlinear strictly elliptic equations are obtained for a suitable growth power of the nonlinearity with respect to the gradient variable (subquadratic for instance). For equations with superquadratic growth power in gradient, one usually uses weak Bernstein-type arguments which require regularity and/or convex-type assumptions on the gradient nonlinearity. In this article, we obtain new Lipschitz regularity results for a large class of nonlinear strictly elliptic equations with possibly arbitrary growth power of the Hamiltonian with respect to the gradient variable using some ideas coming from Ishii-Lions' method. We use these bounds to solve an ergodic problem and to study the regularity and the large time behavior of the solution of the evolution equation.