2014/09/09 by Antoine Gloria, Stefan Neukamm, Gloria, Antoine +3 · 5 citations
Mathematics · #35B65 #35J15 #60H25 #60K37 #Analysis of PDEs (math.AP) #FOS: Mathematics #Probability (math.PR) #math.AP #math.PR #msc:35B65 #msc:35J15 #msc:60H25 #msc:60K37
paper · pdf · doi:10.48550/arxiv.1409.2678
We split the original paper into two parts: regularity theory and quantitative estimates. This part gives a digested version of the regularity theory
arxiv created 2019/10/09 · arxiv updated 2019/10/10
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 Calderéron-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 that r_* 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.