# De Giorgi Techniques Applied to Hamilton-Jacobi Equations with Unbounded Right-Hand Side

## TL;DR

This paper develops uniform Hölder estimates for solutions to second-order Hamilton-Jacobi equations with unbounded sources using De Giorgi's method, extending previous results to more general conditions.

## Contribution

It applies De Giorgi's technique to establish regularity estimates for Hamilton-Jacobi equations with unbounded right-hand sides, a novel approach in this context.

## Key findings

- Established Hölder continuity of solutions under broad conditions
- Achieved estimates uniform in diffusion and Hamiltonian smoothness
- Extended De Giorgi's method to a new class of PDEs

## Abstract

In this article we obtain Holder estimates for solutions to second-order Hamilton-Jacobi equations with super-quadratic growth in the gradient and unbounded source term. The estimates are uniform with respect to the smallness of the diffusion and the smoothness of the Hamiltonian. Our work is in the spirit of a result by P. Cardaliaguet and L. Silvestre [5]. We utilize De Giorgi's method, which was introduced to this class of equations in [6].