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A Regularity Theory for Random Elliptic Operators

par Gloria, Antoine ;Neukamm, Stefan;Otto, Felix
Référence Milan journal of mathematics
Publication Publié, 2020-08-01

Article révisé par les pairs

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Résumé :

Since the seminal results by Avellaneda & Lin it is known that ellipticoperators with periodic coefficients enjoy the same regularity theory as theLaplacian on large scales. In a recent inspiring work, Armstrong & Smart provedlarge-scale Lipschitz estimates for such operators with random coefficients satisfyinga finite-range of dependence assumption. In the present contribution, weextend the intrinsic large-scale regularity of Avellaneda & Lin (namely, intrinsiclarge-scale Schauder and Calderón-Zygmund estimates) to elliptic systems withrandom coefficients. The scale at which this improved regularity kicks in is characterizedby a stationary field r* which we call the minimal radius. This regularitytheory is qualitative in the sense that r* is almost surely finite (which yields a newLiouville theorem) under mere ergodicity, and it is quantifiable in the sense thatr* has high stochastic integrability provided the coefficients satisfy quantitativemixing assumptions. We illustrate this by establishing optimal moment boundson r* for a class of coefficient fields satisfying a multiscale functional inequality,and in particular for Gaussian-type coefficient fields with arbitrary slow-decayingcorrelations.