Fully nonlinear Hamilton-Jacobi equations of degenerate type

TL;DR

This paper studies fully nonlinear degenerate Hamilton-Jacobi equations with superlinear Hamiltonians, proving local Lipschitz regularity of viscosity solutions and applying results to a two-phase free boundary problem.

Contribution

It establishes Lipschitz regularity for viscosity solutions of degenerate Hamilton-Jacobi equations using Ishii-Jensen inequality, with applications to free boundary problems.

Findings

Viscosity solutions are locally Lipschitz continuous.

Estimates depend on structural conditions of the equations.

Application to a two-phase free boundary problem.

Abstract

We examine Hamilton-Jacobi equations driven by fully nonlinear degenerate elliptic operators in the presence of superlinear Hamiltonians. By exploring the Ishii-Jensen inequality, we prove that viscosity solutions are locally Lipschitz-continuous, with estimates depending on the structural conditions of the problem. We close the paper with an application of our findings to a two-phase free boundary problem.