Stochastic homogenization of degenerate integral functionals and their Euler-Lagrange equations

TL;DR

This paper proves stochastic homogenization for degenerate integral functionals with ergodic integrands, establishing convergence of solutions and properties of the homogenized integrand under minimal assumptions.

Contribution

It extends stochastic homogenization results to degenerate integrals with minimal integrability conditions on the integrand and matrix A, including Euler-Lagrange equations.

Findings

Homogenization of degenerate integral functionals under ergodic conditions.

Homogenized integrand retains strict convexity and differentiability.

Convergence results for solutions with boundary or obstacle conditions.

Abstract

We prove stochastic homogenization for integral functionals defined on Sobolev spaces, where the stationary, ergodic integrand satisfies a degenerate growth condition of the form \begin{equation*} c|\xi A(\omega,x)|^p\leq f(\omega,x,\xi)\leq |\xi A(\omega,x)|^p+\Lambda(\omega,x) \end{equation*} for some $p\in (1,+\infty)$ and with a stationary and ergodic diagonal matrix $A$ such that its norm and the norm of its inverse satisfy minimal integrability assumptions. We also consider the convergence when Dirichlet boundary conditions or an obstacle condition are imposed. Assuming the strict convexity and differentiability of $f$ with respect to its last variable, we further prove that the homogenized integrand is also strictly convex and differentiable. These properties allow us to show homogenization of the associated Euler-Lagrange equations.