Homogenization: in Mathematics or Physics?

Homogenization appeared more than 100 years ago. It is an approach to study the macro-behavior of a medium by its micro-properties. In mathematics, homogenization theory considers the limitations of the sequences of the problems and its solutions when a parameter tends to zero. This parameter is regarded as the ratio of the characteristic size in the micro scale to that in the macro scale. So what is considered is a sequence of problems in a fixed domain while the characteristic size in micro scale tends to zero. But for the real situations in physics or engineering, the micro scale of a medium is fixed and can not be changed. In the process of homogenization, it is the size in macro scale which becomes larger and larger and tends to infinity. We observe that the homogenization in physics is not equivalent to the homogenization in mathematics up to some simple rescaling. With some direct error estimates, we explain in what means we can accept the homogenized problem as the limitation of the original real physical problems. As a byproduct, we present some results on the mathematical homogenization of some problems with source term being only weakly compacted in $H^{-1}$, while in standard homogenization theory, the source term is assumed to be at least compacted in $H^{-1}$. A real example is also given to show the validation of our observation and results

Stochastic homogenization of nonconvex unbounded integral functionals with convex growth

We consider the well-travelled problem of homogenization of random integral functionals. When the integrand has standard growth conditions, the qualitative theory is well-understood. When it comes to unbounded functionals, that is, when the domain of the integrand is not the whole space and may depend on the space-variable, there is no satisfactory theory. In this contribution we develop a complete qualitative stochastic homogenization theory for nonconvex unbounded functionals with convex growth. We first prove that if the integrand is convex and has $p$-growth from below (with $p>d$, the dimension), then it admits homogenization regardless of growth conditions from above. This result, that crucially relies on the existence and sublinearity at infinity of correctors, is also new in the periodic case. In the case of nonconvex integrands, we prove that a similar homogenization result holds provided the nonconvex integrand admits a two-sided estimate by a convex integrand (the domain of which may depend on the space-variable) that itself admits homogenization. This result is of interest to the rigorous derivation of rubber elasticity from polymer physics, which involves the stochastic homogenization of such unbounded functionals.Comment: 64 pages, 2 figure

On cell problems for Hamilton-Jacobi equations with non-coercive Hamiltonians and its application to homogenization problems

We study a cell problem arising in homogenization for a Hamilton-Jacobi equation whose Hamiltonian is not coercive. We introduce a generalized notion of effective Hamiltonians by approximating the equation and characterize the solvability of the cell problem in terms of the generalized effective Hamiltonian. Under some sufficient conditions, the result is applied to the associated homogenization problem. We also show that homogenization for non-coercive equations fails in general

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