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

Ruf, Matthias; Ruf, Thomas

Journal de l’École polytechnique — Mathématiques, Tome 10 (2023), pp. 253-303

We prove stochastic homogenization for integral functionals defined on Sobolev spaces, where the stationary, ergodic integrand satisfies a degenerate growth condition of the form ${c\left|\xi A(\omega ,x)\right|}^p\le f(\omega ,x,\xi )\le {\left|\xi A(\omega ,x)\right|}^p+\Lambda (\omega ,x)$ 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 and $\Lambda$ is a nonnegative, stationary function with finite first moment. 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.

Nous prouvons l’homogénéisation stochastique des fonctionnelles intégrales définies sur des espaces de Sobolev, où l’intégrande stationnaire et ergodique satisfait à une condition de croissance dégénérée de la forme ${c\left|\xi A(\omega ,x)\right|}^p\le f(\omega ,x,\xi )\le {\left|\xi A(\omega ,x)\right|}^p+\Lambda (\omega ,x)$ où $p\in (1,+\infty$), $A$ est une matrice diagonale stationnaire et ergodique dont la norme et celle de son inverse satisfont aux hypothèses d’intégrabilité minimale et $\Lambda$ est une fonction stationnaire et non négative avec un moment d’ordre un fini. Nous considérons également la convergence lorsque des conditions aux limites de Dirichlet ou une condition d’obstacle sont imposées. En supposant la stricte convexité et la différentiabilité de $f$ par rapport à sa dernière variable, nous prouvons en outre que l’intégrande homogénéisée est également strictement convexe et différentiable. Ces propriétés nous permettent de montrer l’homogénéisation des équations d’Euler-Lagrange associées.

Classification : 49J45, 49J55, 60G10, 35J70
Keywords: Stochastic homogenization, integral functionals, degenerate $p$-growth, homogenization of Euler-Lagrange equations
Mots-clés : Homogénéisation stochastique, fonctionnelles intégrales, croissance d’ordre $p$ dégénérée, homogénéisation des équations d’Euler-Lagrange

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