H\"older regularity for viscosity solutions of fully nonlinear, local or nonlocal, Hamilton-Jacobi equations with super-quadratic growth in the gradient

TL;DR

This paper proves Hölder continuity of viscosity solutions for a class of fully nonlinear Hamilton-Jacobi equations with super-quadratic gradient growth, using representation formulas and weak reverse inequalities.

Contribution

It establishes Hölder regularity for solutions of local and nonlocal Hamilton-Jacobi equations with super-quadratic growth, a result not previously known.

Findings

Viscosity solutions are Hölder continuous with a growth-dependent modulus.

Representation formulas relate nonlocal equations to controlled jump processes.

A weak reverse inequality is used to establish regularity.

Abstract

Viscosity solutions of fully nonlinear, local or non local, Hamilton-Jacobi equations with a super-quadratic growth in the gradient variable are proved to be H\"older continuous, with a modulus depending only on the growth of the Hamiltonian. The proof involves some representation formula for nonlocal Hamilton-Jacobi equations in terms of controlled jump processes and a weak reverse inequality.