Quantitative estimates in stochastic homogenization for correlated coefficient fields

Quantitative theory in stochastic homogenization

This course relies on a work in preparation with A. Gloria [2], which is a continuum version of [1]. It slightly differs from [2] because the present analysis does not rely on Green’s functions and treats the periodic case. For related work on an emerging quantitative theory of stochastic homogenization, including many references, we refer to three preprints, which are available on my web page:...

On Generalized Gaussian Free Fields and Stochastic Homogenization

We study a generalization of the notion of Gaussian free field (GFF). Although the extension seems minor, we first show that a generalized GFF does not satisfy the spatial Markov property, unless it is a classical GFF. In stochastic homogenization, the scaling limit of the corrector is a possibly generalized GFF described in terms of an “effective fluctuation tensor” that we denote by Q. We pro...

Coefficient Estimates for Univalent Functions

Coefficient estimates for Ruscheweyh derivatives

are classes of starlike and strongly starlike functions of order β (0 < β ≤ 1), respectively. Note that S∗(β)⊂ S∗ for 0< β< 1 and S∗(1)= S∗ [5]. Kanas [2] introduced the subclass R̄δ(β) of function f ∈ S as the following. Definition 1.1. For δ ≥ 0, β ∈ (0,1], a function f normalized by (1.1) belongs to R̄δ(β) if, for z ∈D−{0} and Dδf(z)≠ 0, the following holds: ∣∣∣arg z ( Dδf(z) )′ Dδf(z) ∣∣∣≤ βπ...

Uniform Estimates in Two Periodic Homogenization Problems

We analyze two partial diierential equations that are posed on perforated domains. We provide a priori estimates, that do not depend on the size of the perforation: a sequence of solutions is uniformly bounded in a Sobolev space of regular functions. The rst homogeniza-tion problem concerns the Laplace-and the mean-curvature operator with Neumann boundary conditions. We derive uniform Lipschitz...