期刊
MILAN JOURNAL OF MATHEMATICS
卷 88, 期 1, 页码 99-170出版社
SPRINGER BASEL AG
DOI: 10.1007/s00032-020-00309-4
关键词
Quantitative stochastic homogenization; large scale regularity
资金
- European Research Council under the European Community's Seventh Framework Programme (FP7/2014-2019 Grant) [QUANTHOM 335410]
- DFG
- Chaire Schlumberger at IHES
Since the seminal results by Avellaneda & Lin it is known that elliptic operators with periodic coefficients enjoy the same regularity theory as the Laplacian on large scales. In a recent inspiring work, Armstrong & Smart proved large-scale Lipschitz estimates for such operators with random coefficients satisfying a finite-range of dependence assumption. In the present contribution, we extend the intrinsic large-scale regularity of Avellaneda & Lin (namely, intrinsic large-scale Schauder and Calderon-Zygmund estimates) to elliptic systems with random coefficients. The scale at which this improved regularity kicks in is characterized by a stationary field r(*) which we call the minimal radius. This regularity theory is qualitative in the sense that r(*) is almost surely finite (which yields a new Liouville theorem) under mere ergodicity, and it is quantifiable in the sense thatr(*) has high stochastic integrability provided the coefficients satisfy quantitative mixing assumptions. We illustrate this by establishing optimal moment bounds on r(*) for a class of coefficient fields satisfying a multiscale functional inequality, and in particular for Gaussian-type coefficient fields with arbitrary slow-decaying correlations.
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