New homogenization results for convex integral functionals and their Euler–Lagrange equations

We study stochastic homogenization for convex integral functionals u↦∫DW(ω,xε,∇u)dx,whereu:D⊂Rd→Rm,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} u\mapsto \int _D W(\omega ,\tfrac{x}{\varepsilon },\nabla u)\,\textrm{d}x,\quad \text{ where }\quad u:D\subset {\mathbb {R}}^d\rightarrow {\mathbb {R}}^m, \end{aligned}$$\end{document}defined on Sobolev spaces. Assuming only stochastic integrability of the map ω↦W(ω,0,ξ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \mapsto W(\omega ,0,\xi )$$\end{document}, we prove homogenization results under two different sets of assumptions, namely ∙1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bullet _1$$\end{document}W satisfies superlinear growth quantified by the stochastic integrability of the Fenchel conjugate W∗(·,0,ξ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W^*(\cdot ,0,\xi )$$\end{document} and a certain monotonicity condition that ensures that the functional does not increase too much by componentwise truncation of u,∙2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bullet _2$$\end{document}W is p-coercive in the sense |ξ|p≤W(ω,x,ξ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\xi |^p\le W(\omega ,x,\xi )$$\end{document} for some p>d-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p>d-1$$\end{document}.