An introduction to the qualitative and quantitative theory of homogenization

We present an introduction to periodic and stochastic homogenization of ellip- tic partial differential equations. The first part is concerned with the qualitative theory, which we present for equations with periodic and random coefficients in a unified approach based on Tartar's method of oscillating test functions. In partic- ular, we present a self-contained and elementary argument for the construction of the sublinear corrector of stochastic homogenization. (The argument also applies to elliptic systems and in particular to linear elasticity). In the second part we briefly discuss the representation of the homogenization error by means of a two- scale expansion. In the last part we discuss some results of quantitative stochastic homogenization in a discrete setting. In particular, we discuss the quantification of ergodicity via concentration inequalities, and we illustrate that the latter in combi- nation with elliptic regularity theory leads to a quantification of the growth of the sublinear corrector and the homogenization error.Comment: Lecture notes of a minicourse given by the author during the GSIS International Winter School 2017 on "Stochastic Homogenization and its applications" at the Tohoku University, Sendai, Japan; This version contains a correction of Lemma 2.1

Homogenization of the nonlinear bending theory for plates

We carry out the spatially periodic homogenization of nonlinear bending theory for plates. The derivation is rigorous in the sense of Gamma-convergence. In contrast to what one naturally would expect, our result shows that the limiting functional is not simply a quadratic functional of the second fundamental form of the deformed plate as it is the case in nonlinear plate theory. It turns out that the limiting functional discriminates between whether the deformed plate is locally shaped like a "cylinder" or not. For the derivation we investigate the oscillatory behavior of sequences of second fundamental forms associated with isometric immersions, using two-scale convergence. This is a non-trivial task, since one has to treat two-scale convergence in connection with a nonlinear differential constraint.Comment: 36 pages, 4 figures. Major revisions of Sections 2,3 and 4. In Section 2: Correction of definition of conical and cylindrical part (Definition 1). In Section 3: Modifications in the proof of Proposition 2 due to changes in Definition 1. Several new lemmas and other modifications. In Section 4: Modification of proof of lower bound. Proof of upper bound completely revised. Several lemmas adde

Stochastic homogenization of Λ\Lambda-convex gradient flows

In this paper we present a stochastic homogenization result for a class of Hilbert space evolutionary gradient systems driven by a quadratic dissipation potential and a $\Lambda$-convex energy functional featuring random and rapidly oscillating coefficients. Specific examples included in the result are Allen-Cahn type equations and evolutionary equations driven by the $p$-Laplace operator with $p\in (1,\infty)$. The homogenization procedure we apply is based on a stochastic two-scale convergence approach. In particular, we define a stochastic unfolding operator which can be considered as a random counterpart of the well-established notion of periodic unfolding. The stochastic unfolding procedure grants a very convenient method for homogenization problems defined in terms of ($\Lambda$-)convex functionals.Comment: arXiv admin note: text overlap with arXiv:1805.0954

H-compactness of elliptic operators on weighted Riemannian Manifolds

In this paper we study the asymptotic behavior of second-order uniformly elliptic operators on weighted Riemannian manifolds. They naturally emerge when studying spectral properties of the Laplace-Beltrami operator on families of manifolds with rapidly oscillating metrics. We appeal to the notion of H-convergence introduced by Murat and Tartar. In our main result we establish an H-compactness result that applies to elliptic operators with measurable, uniformly elliptic coefficients on weighted Riemannian manifolds. We further discuss the special case of ``locally periodic'' coefficients and study the asymptotic spectral behavior of compact submanifolds of $\mathbb R^n$ with rapidly oscillating geometry.Comment: Major revision: In particular, we added various examples and visualization