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

M. Ruf, T. Ruf
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引用次数: 5

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 c|ξA(ω, x)| ≤ f(ω, x, ξ) ≤ |ξA(ω, x)| +Λ(ω, x) for some p ∈ (1,+∞) 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.
退化积分泛函的随机均匀化及其欧拉-拉格朗日方程
我们证明了Sobolev空间上定义的积分泛函的随机齐次化,其中平稳遍历被积函数满足退化生长条件c|ξ a (ω, x)|≤f(ω, x, ξ)≤|ξ a (ω, x)| +Λ(ω, x),对于某p∈(1,+∞),并且具有平稳遍历对角矩阵a,使得其范数及其逆范数满足最小可积假设。我们还考虑了狄利克雷边界条件和障碍条件下的收敛性。假设f对其最后一个变量具有严格的凸性和可微性,进一步证明了齐次被积函数也是严格凸可微的。这些性质使我们能够证明相关欧拉-拉格朗日方程的均匀性。
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