Introduction to Homogenization

Related notions
Variational techniques, multiscale models

Notions that will be studied
We will study the classical elliptic equation in divergence form: $$-\mathrm{d}\mathrm{i}\mathrm{v}(a\left(\frac{x}{\epsilon }\right)\mathrm{\nabla }{u}^{\epsilon }(x))=f(x),$$ where $\epsilon \ll 1$ is the small scale, and where $a$ is a ${\mathbb{Z}}^d$-periodic coefficient field. This equation is relevant for modeling various physical phenomena (e.g. thermal or mechanical equilibrium, electrostatics) in multiscale materials.


Figure 1: Example of coefficient field $a(x/\epsilon )$ on (a), right-hand side $f$ on (b), solution $u^{\epsilon }$ on (c), and its derivative ${\partial }_1{u}^{\epsilon }$ on (d).

We show that an averaging process occurs when the small $\epsilon$ vanishes and that the solution $u^{\epsilon }$ to (1) can be approximated by the (simpler) solution of the homogenized problem $$-\mathrm{d}\mathrm{i}\mathrm{v}(\bar{a}\mathrm{\nabla }\bar{u}(x))=f(x),$$ where $\bar{a}$ is a constant matrix. The oscillating gradient $\mathrm{\nabla }{u}^{\epsilon }$ is then retrieved by means of the two-scale expansion.

In this regard, the following mathematical notions will be under our scope:

• the Lax-Milgram theorem and the Fredholm alternative,
• the correctors and the two-scale expansion,
• the div-curl lemma, Tartar's method and the H-convergence,
• the Hashin-Shtrikman bounds.

• G. Allaire.Shape optimization by the homogenization method, volume 146 of Applied Mathematical Sciences. Springer-Verlag, New York, 2002.
• H. Brezis. Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York, 2011.
• V. Jikov, S. Kozlov, and O. Oleinik. Homogenization of differential operators and integral functionals. Springer-Verlag, Berlin, 1994.
• L. Tartar. The general theory of homogenization, volume 7 of Lecture Notes of the Unione Matematica Italiana. Springer-Verlag, Berlin; UMI, Bologna, 2009.

Date and time info
Tuesday, 9h15-10h45

Keywords
PDEs, elliptic theory, multiscale

Prerequisites
Courses PDE 1 & 2, Basic functional analysis H¹ spaces, compactness, weak convergence), basics of Partial Differential Equations (elliptic theory, weak solution)