Title: $Γ$-convergence and stochastic homogenization for functionals in the $\mathcal{A}$-free setting Show Chinese title

Title: $Γ$-收敛和在$\mathcal{A}$-自由设置中的泛函随机均质化

Abstract: We obtain a compactness result for $\Gamma$-convergence of integral functionals defined on $\mathcal{A}$-free vector fields. This is used to study homogenization problems for these functionals without periodicity assumptions. More precisely, we prove that the homogenized integrand can be obtained by taking limits of minimum values of suitable minimization problems on large cubes, when the side length of these cubes tends to $+\infty$, assuming that these limit values do not depend on the center of the cube. Under the usual stochastic periodicity assumptions, this result is then used to solve the stochastic homogenization problem by means of the subadditive ergodic theorem. Show Chinese abstract

Abstract: 我们得到了关于在$\mathcal{A}$-自由向量场上定义的积分泛函的$\Gamma$-收敛性的紧性结果。 这是用来研究这些泛函的均质化问题，而无需周期性假设。 更准确地说，我们证明了当这些立方体的边长趋于$+\infty$时，通过取大立方体上适当极小化问题的极小值的极限，可以得到均质化的被积函数，假设这些极限值不依赖于立方体的中心。 在通常的随机周期性假设下，这个结果随后被用来通过次可加遍历定理解决随机均质化问题。