WIAS Preprint No. 2971, (2022)

Regularity of random elliptic operators with degenerate coefficients and applications to stochastic homogenization


Authors

  • Bella, Peter
    ORCID: 0000-0002-1660-1711
  • Kniely, Michael
    ORCID: 0000-0001-5645-4333

2020 Mathematics Subject Classification

  • 35J70 35R60 35B65 35B27

Keywords

  • Degenerate elliptic equations, random coefficients, large-scale regularity, stochastic homogenization, exponential moment bounds, sensitivity estimates

DOI

10.20347/WIAS.PREPRINT.2971

Abstract

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 regularity of a-harmonic functions in a degenerate situation, we provide stretched exponential moments for the minimal radius describing the minimal scale for this 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 the coefficient field and its inverse. We also introduce the ellipticity radius, which encodes the minimal scale where these moments are close to their positive expectation value.

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