A note on the regularity of solutions of Hamilton-Jacobi equations with superlinear growth in the gradient variable

TL;DR

This paper proves that solutions to certain Hamilton-Jacobi equations with superlinear gradient growth are locally Hölder continuous, with the regularity depending only on the growth rate, using a reverse Hölder inequality.

Contribution

It establishes the local Hölder continuity of solutions for Hamilton-Jacobi equations with superlinear growth, providing a new regularity result based on a reverse Hölder inequality.

Findings

Solutions are locally Hölder continuous

Hölder exponent depends only on Hamiltonian growth

Proof uses reverse Hölder inequality

Abstract

We investigate the regularity of solutions of first order Hamilton-Jacobi equation with super linear growth in the gradient variable. We show that the solutions are locally H\"older continuous with H\"older exponent depending only on the growth of the Hamiltonian. The proof relies on a reverse H\"older inequality.