Calculus of Variations and Geometric Measure Theory

N. Dirr - F. Dragoni - P. Mannucci - C. Marchi

Stochastic homogenization for functionals with anisotropic rescaling and non-coercive Hamilton-Jacobi equations

created by dragoni on 18 Jul 2017
modified on 02 Jul 2019


Published Paper

Inserted: 18 jul 2017
Last Updated: 2 jul 2019

Journal: SIAM Journal on Mathematical Analysis
Volume: 50
Number: 5
Pages: 5198-5242
Year: 2018


We study the stochastic homogenization for a Cauchy problem for a first-order Hamilton-Jacobi equation whose operator is not coercive w.r.t. the gradient variable. We look at Hamiltonians like $H(x,\sigma(x)p,\omega)$ where $\sigma(x)$ is a matrix associated to a Carnot group. The rescaling considered is consistent with the underlying Carnot group structure, thus anisotropic. We will prove that under suitable assumptions for the Hamiltonian, the solutions of the $\varepsilon$-problem converge to a deterministic function which can be characterized as the unique (viscosity) solution of a suitable deterministic Hamilton-Jacobi problem.

Keywords: Stochastic homogenisation, Carnot groups, non-coercive Hamiltonian