# Perturbation problems in homogenization of hamilton-jacobi equations

## TL;DR

This paper investigates how the ergodic constant in convex Hamilton-Jacobi equations behaves under periodic and random perturbations, revealing dimension-dependent first-order effects and extending nonlinear homogenization theory.

## Contribution

It provides the first first-order Taylor expansion of the ergodic constant in nonlinear Hamilton-Jacobi homogenization, highlighting dimension-dependent phenomena.

## Key findings

- First-order term is non-trivial in dimension 1.
- For dimensions ≥ 2, the first-order term vanishes.
- Results extend homogenization theory to nonlinear Hamilton-Jacobi equations.

## Abstract

This paper is concerned with the behavior of the ergodic constant associated with convex and superlinear Hamilton-Jacobi equation in a periodic environment which is perturbed either by medium with increasing period or by a random Bernoulli perturbation with small parameter. We find a first order Taylor's expansion for the ergodic constant which depends on the dimension d. When d = 1 the first order term is non trivial, while for all d $\ge$ 2 it is always 0. Although such questions have been looked at in the context of linear uniformly elliptic homogenization, our results are the first of this kind in nonlinear settings. Our arguments, which rely on viscosity solutions and the weak KAM theory, also raise several new and challenging questions.