

Introduction to Homogenization
- 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 modelsNotions 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.
Regular lectures: Winter semester 2019/2020
- Topics in the Regularity Theory for Elliptic and Parabolic PDE's
- Renan Assimos Martins, Jonas Hirsch, Aleksander Klimek, Konstantinos Zemas
- Wednesdays, 15-17, MPI MiS A3 02
- Littlewood-Paley theory in PDEs
- Jan Burczak, Jonas Hirsch
- Thursdays, 15:15- 16:45, Leipzig University, Augusteum, A-314
- Seven Lectures in Algebraic Statistics
- Eliana Duarte, Orlando Marigliano
- Monday 13:00 to 14:30, starting October 21, MPI MiS G3 10
- Introduction to stochastic thin film equations
- Benjamin Gess
- Mo, 16:15-17:45, MPI MiS A3 01
- Introduction to game theory
- Jules Hedges
- Tuesday, 11.15-12.45, MPI MiS A3 02
- Introduction to Homogenization
- Marc Josien
- Tuesday, 9h15-10h45, MPI MiS A3 02
- Mathematical Topics in Neuroscience
- Jürgen Jost
- Fr 1.30pm, MPI MiS A3 01
- Nonlinear Algebra
- Mateusz Michałek
- Monday, 9.30, MPI MiS G3 10
- Logarithmic Sobolev Inequality
- Felix Otto
- Thursday, 09.15-11.00, MPI MiS A3 01
- Optimierung I
- André Uschmajew
- Tue 15:15 - 16:45, Wed 15:15 - 16:45, Leipzig University, Hs 5 (Tue), Hs 19 (Wed)
- Ringvorlesung
- IMPRS Faculty, Daniela Cadamuro, Angkana Rüland, Matteo Smerlak
- Friday 09.15-11.00 (Rüland) 09.30-11.00 (Cadamuro) 09.15-11.00 (Smerlak) Exercice classes: Friday 11.00-12.00, MPI MiS G3 10