Lipschitz Regularity for Elliptic Equations with Random Coefficients

By S. Armstrong, Jean-Christophe Mourrat. 2014

128 citations recorded by the source corpus.

We develop a higher regularity theory for general quasilinear elliptic equations and systems in divergence form with random coefficients. The main result is a large-scale L∞-type estimate for the gradient of a solution. The estimate is proved with optimal stochastic integrability under a one-parameter family of mixing assumptions, allowing for very weak mixing with non-integrable correlations to very strong mixing (for example finite range of dependence). We also prove a quenched L2 estimate for the error in homogenization of Dirichlet problems. The approach is based on subadditive arguments which rely on a variational formulation of general quasilinear divergence-form equations.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Finance

View Paper: https://www.semanticscholar.org/paper/cc1e9cb583eef5ab3622b83c528a95e99dfdde7a

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In Applied Mathematics

In Computational Theory and Mathematics

In Finance

Citations in the corpus

Boundary Estimates in Elliptic Homogenization (2015, methodology): “This allows us to use a general scheme for establishing Lipschitz estimates down to the scale ε, which was formulated recently in [4] and used for interior estimates in stochastic homogenization with random coefficients (also see [2, 1] as well as related work in [21, 22, 20, 19]).”
A higher-order large-scale regularity theory for random elliptic operators (2015, unknown): “In the context of nonlinear elliptic systems in divergence form, the result of Armstrong and Smart [3] on the large-scale C0,1-estimate was generalized by Armstrong and Mourrat [2] to nonsymmetric coe cients and well beyond nite range, further con rming that the random large-scale regularity theory ”
Quantitative Stochastic Homogenization and Large-Scale Regularity (2017, background): “The arguments in this chapter follows these ideas, although the presentation here is new and much simpler than that of [13].”
Mesoscopic Higher Regularity and Subadditivity in Elliptic Homogenization (2015, methodology): “This is because, as in [4, 2], our method allows us to use subadditivity to control the stochastic fluctuations.”
A Regularity Theory for Random Elliptic Operators (2014, methodology): “Compared to the series [4, 6, 7], our approach has advantages both in terms of stochastic results and large-scale regularity.”
Quantitative estimates in stochastic homogenization for correlated coefficient fields (2019, unknown): “We refer the reader to the companion article [21] for a thorough discussion of the literature, and in particular the works [30, 17] based on functional inequalities, and the work of [5, 4] based on finite-range or α-mixing assumptions.”
Quantitative stochastic homogenization and regularity theory of parabolic equations (2017, methodology): “In this section, we obtain an algebraic rate of convergence for the limits of the subadditive quantities by adapting the approach of [6, 5], following the presentation of [2, Chapter 2].”
The additive structure of elliptic homogenization (2016, methodology): “They are essentially a rephrasing of the results of [4, 3], which give an algebraic rate of convergence for certain subadditive energy quantities that are close to J .”