Sobolev regularity for the first order Hamilton-Jacobi equation

TL;DR

This paper establishes Sobolev regularity and almost everywhere differentiability for solutions of first order Hamilton-Jacobi equations with superlinear Hamiltonians, using inverse H"older inequalities, with applications to mean field games.

Contribution

It provides new Sobolev estimates and differentiability results for Hamilton-Jacobi equations with superlinear Hamiltonians, advancing regularity theory in this area.

Findings

Solutions are in Sobolev spaces with specific estimates.

Solutions are differentiable almost everywhere.

Applications demonstrated in mean field game contexts.

Abstract

We provide Sobolev estimates for solutions of first order Hamilton-Jacobi equations with Hamiltonians which are superlinear in the gradient variable. We also show that the solutions are differentiable almost everywhere. The proof relies on an inverse H\"older inequality. Applications to mean field games are discussed.