Introduction to Homogenization

Related notions
Variational techniques, multiscale models

Notions that will be studied
We will study the classical elliptic equation in divergence form: $$ -{\rm div}\left( a\left(\frac{x}{\varepsilon}\right) \nabla u^\varepsilon(x)\right) = f(x), $$ where \(\varepsilon \ll 1\) 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 \(a(x/\varepsilon)\) on (a), right-hand side \(f\) on (b), solution \(u^\varepsilon\) on (c), and its derivative \(\partial_1 u^\varepsilon\) on (d).

We show that an averaging process occurs when the small \(\varepsilon\) vanishes and that the solution \(u^\varepsilon\) to (1) can be approximated by the (simpler) solution of the homogenized problem $$ -{\rm div}\left( \overline{a} \nabla \overline{u}(x)\right) = f(x), $$ where \(\overline{a}\) is a constant matrix. The oscillating gradient \(\nabla u^\varepsilon\) 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)