• Lecturer: Marc Josien
• Date: Tuesday, 9h15-10h45
• Room: MPI MiS A3 02
• 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)

Abstract

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.

References:
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.