论文标题
随机椭圆算子的规律性，具有退化系数和应用于随机均质化
Regularity of random elliptic operators with degenerate coefficients and applications to stochastic homogenization
论文作者
论文摘要
我们考虑以对称的随机系数字段$ a $ $ a $的形式考虑二阶椭圆方程。扩展了第一作者Fehrman和Otto的作品[Ann。应用。概率。 28（2018），没有。 3，1379-1422]，他建立了大规模的$ c^{1，α} $ $ a $ a harmonic函数的规律性，我们在退化的情况下为最小的半径$ r _*$提供了伸展的指数矩，描述了这种$ c^{1，α} $ juromist的最低规模。作为随机均质化的应用，我们部分概括了Gloria，Neukamm和Otto的结果[Anal。 PDE 14（2021），否。 8，2497-2537]关于校正器的生长，其梯度的衰减以及对退化设置的定量两尺度扩展。在技术层面上，我们要求系数集合固定并遵守频谱差距不平等，并且在$ a $ a $ a $ a^{ - 1} $上强加了矩界。我们还介绍了椭圆度半径$ r_e $，该半径$ r_e $编码这些时刻接近其积极期望值的最低规模。
We consider degenerate elliptic equations of second order in divergence form with a symmetric random coefficient field $a$. Extending the work of the first author, Fehrman, and Otto [Ann. Appl. Probab. 28 (2018), no. 3, 1379-1422], who established the large-scale $C^{1,α}$ regularity of $a$-harmonic functions in a degenerate situation, we provide stretched exponential moments for the minimal radius $r_*$ describing the minimal scale for this $C^{1,α}$ regularity. As an application to stochastic homogenization, we partially generalize results by Gloria, Neukamm, and Otto [Anal. PDE 14 (2021), no. 8, 2497-2537] on the growth of the corrector, the decay of its gradient, and a quantitative two-scale expansion to the degenerate setting. On a technical level, we demand the ensemble of coefficient fields to be stationary and subject to a spectral gap inequality, and we impose moment bounds on $a$ and $a^{-1}$. We also introduce the ellipticity radius $r_e$ which encodes the minimal scale where these moments are close to their positive expectation value.
