Homogenization of PDEs

Introduction

Recommended prerequisites: Functional Analysis (obligatory) and elementary theory of PDEs. We will cover some necessary background materials at the beginning of the course.

The course is devoted to an introduction to the mathematical theory of homogenization for partial differential equations, namely some basic techniques for studying PDEs with rapidly oscillating coefficients. PDEs with rapidly oscillating coefficients appear in various applications such as electricity and magnetism in heterogeneous media, flows in porous media or thermoregulation phenomena in the human microvascular system, etc. The goal of this theory is to characterize properties of material or system behavior based on investigating the properties of its microscopic structure or elementary processes.  This course will focus mainly on the context of periodic homogenization. However, we would like to emphasize that homogenization is not restricted to the periodic case, but can be applied to any kind of disordered media.

We first cover quickly some necessary background materials, namely L^p-spaces, Sobolev spaces, theory of weak convergence in Banach space,  the compactness theorems, the Lax-Milgram theorem... Next we introduce periodic homogenization theory via some basic tools:  the two-scale asymptotic expansion, the oscillating test function method, the two-scale convergence method and the periodic unfolding method. Finally, we illustrate the methods via the study of some models of periodically microstructured materials arising in applications, which are described by PDEs with fast varying coefficients. We study the limiting process of the solution as the size of the microstructure tends to 0 and also the corrector result for the solution.

This course is addressed to all graduate students in mathematics and engineering sciences.

Lecturer

Lecturer: Dr. Kim-Hang Le

Extent

  • ECTS credits: 4
  • Hours per week: 2+1

Times and rooms

  • Lecture (starting on 30.04.19)
    • Tuesday 10:0012:00:         N24, 131
  • Exercises (every two weeks)
    • Do 16:0018:00:          N24, 131

Literature

[1]   D. Cioranescu, P. Donato, An Introduction to Homogenization, Oxford Lecture Series in Mathematics and Its Applications, 2000.

[2]   G. Allaire, Shape optimization by the homogenization method App. Math. Sci., NY, Springer, 2002.

[3]  A. Bensoussan, J.-L. Lions, G. Papanicolaou, Asymptotic Analysis for Periodic Structures, North Holland, Amsterdam, 1978.

[4] O.A. Oleinik, A.S. Schamaev and G.A. Yosifian, Mathematical Problems in Elasticity and Homogenization, North-Holland, Amsterdam, London, New York, Tokyo, 1992.

 Link to the Semesterapparat.