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Quantitative
estimates in stochastic homogenization for correlated
coefficient fields
Antoine Gloria, Stefan Neukamm and Felix Otto |
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Vol. 14 (2021), No. 8, 2497–2537
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Abstract |
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This paper is about the homogenization of linear elliptic operators in divergence form with stationary random coefficients that have only slowly decaying correlations. It deduces optimal estimates of the homogenization error from optimal growth estimates of the (extended) corrector. In line with the heuristics, there are transitions at dimension , and for a correlation-decay exponent we capture the correct power of logarithms coming from these two sources of criticality. The decay of correlations is sharply encoded in terms of a multiscale logarithmic Sobolev inequality (LSI) for the ensemble under consideration — the results would fail if correlation decay were encoded in terms of an -mixing condition. Among other ensembles popular in modeling of random media, this class includes coefficient fields that are local transformations of stationary Gaussian fields. The optimal growth of the corrector is derived from bounding the size of spatial averages of its gradient. This in turn is done by a (deterministic) sensitivity estimate of , that is, by estimating the functional derivative of with respect to the coefficient field . Appealing to the LSI in form of concentration of measure yields a stochastic estimate on . The sensitivity argument relies on a large-scale Schauder theory for the heterogeneous elliptic operator . The treatment allows for nonsymmetric and for systems like linear elasticity. |
Keywords
stochastic homogenization, convergence rates, fat tails
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Mathematical Subject Classification 2010
Primary: 35J15, 35J47, 60H25, 74Q05
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Milestones
Received: 11 October 2019
Revised: 8 May 2020
Accepted: 16 June 2020
Published: 19 December 2021
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Authors |
| Antoine Gloria | ||
| Sorbonne Université, CNRS Université de Paris Laboratoire Jacques-Louis Lions Paris France |
Institut Universitaire de France | Université Libre de Bruxelles Département de Mathématique Brussels Belgium |
| Stefan Neukamm | ||
| Faculty of Mathematics Technische Universität Dresden Dresden Germany |
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| Felix Otto | ||
| Max Planck Institute for Mathematics
in the Sciences Leipzig Germany |
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