Mathematics > Analysis of PDEs
[Submitted on 3 Oct 2022 (v1), last revised 4 Dec 2023 (this version, v2)]
Title:Regularity of random elliptic operators with degenerate coefficients and applications to stochastic homogenization
View PDFAbstract: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 $C^{1,\alpha}$ 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 $C^{1,\alpha}$ 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.
Submission history
From: Michael Kniely [view email][v1] Mon, 3 Oct 2022 19:12:04 UTC (33 KB)
[v2] Mon, 4 Dec 2023 23:04:25 UTC (36 KB)
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