$\mathbb{A}$-弱可微映射空间中非凸泛函的随机均匀化
Stochastic Homogenisation of nonconvex functionals in the space of $\mathbb{A}$-weakly differentiable maps
AI总结：
本文证明了具有有限维零空间的向量微分算子$\mathbb{A}$类下随机积分泛函序列的$Γ$-收敛，推广了全梯度结果，并通过blow-up方法变体与Akcloglu-Krengel次加性遍历定理实现了均匀化。
摘要：
我们证明了形如 \begin{equation*} \int_{U} f\Big(ω, x/\varepsilon, \mathbb{A} u\Big) \mathrm{d} x \end{equation*} 的微分约束随机积分泛函序列的 $Γ$-收敛，适用于具有有限维零空间的向量微分算子 $\mathbb{A}$ 类。本研究旨在推广全梯度的结果，并涵盖对称梯度和偏量算子的情形。均匀化过程通过在 $\mathbb{A}$-弱可微映射设定中采用 blow-up 方法变体，并结合 Akcloglu-Krengel 次加性遍历定理来实现。
英文摘要：
We prove the $Γ$-convergence of sequences of differentially constrained, random integral functionals of the form \begin{equation*} \int_{U} f\Big(ω, x/\varepsilon, \mathbb{A} u\Big) \mathrm{d} x \end{equation*} for the class of vectorial differential operators $\mathbb{A}$ with finite-dimensional nullspaces. This work is intended to generalise results for the full gradient and to cover the cases of symmetric gradients and the deviatoric operator. The homogenisation procedure is carried out by employing a variant of the blow-up method in the setting of $\mathbb{A}$-weakly differentiable maps along with the Akcloglu-Krengel subadditive ergodic theorem.
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