H\"older estimates in space-time for viscosity solutions of Hamilton-Jacobi equations

TL;DR

This paper establishes uniform H"older continuity estimates in space and time for viscosity solutions of Hamilton-Jacobi equations, including degenerate parabolic cases, addressing the lack of Lipschitz regularity.

Contribution

It provides the first known uniform H"older estimates for viscosity solutions of Hamilton-Jacobi equations under minimal regularity assumptions.

Findings

Solutions are H"older continuous in space and time.

Uniform estimates hold for degenerate parabolic equations.

Results apply to Hamiltonians with superlinear growth at infinity.

Abstract

It is well-known that solutions to the basic problem in the calculus of variations may fail to be Lipschitz continuous when the Lagrangian depends on t. Similarly, for viscosity solutions to time-dependent Hamilton-Jacobi equations one cannot expect Lipschitz bounds to hold uniformly with respect to the regularity of coefficients. This phenomenon raises the question whether such solutions satisfy uniform estimates in some weaker norm. We will show that this is the case for a suitable H\"older norm, obtaining uniform estimates in (x,t) for solutions to first and second order Hamilton-Jacobi equations. Our results apply to degenerate parabolic equations and require superlinear growth at infinity, in the gradient variables, of the Hamiltonian. Proofs are based on comparison arguments and representation formulas for viscosity solutions, as well as weak reverse H\"older inequalities.