Type of course

Obligatory prerequisites

Course content

The objective of the course is to introduce homogenization theory for partial differential equations and integral functionals.

Composite materials are widely used in industry and include such well-known examples as superconductors and optical fibers. However, modelling these materials is difficult, since they often have different properties at different points. The mathematical theory of homogenization is designed to handle this problem. The theory uses an idealized homogenous material to model a real composite while taking into account the microscopic structure. This introduction to homogenization theory develops the natural framework of the theory regarding the variational methods for partial differential equations. We discuss the homogenization of several kinds of second-order boundary value problems. We focus separately on classical examples of steady and non-steady heat equations, the wave equation, and the linearized system of elasticity. Numerous illustrations and examples are given.

Weak and weak* convergence in Banach spaces, rapidly oscillating periodic functions, some classes of Sobolev function, some variational elliptic problems, examples of periodic composite materials, homogenization of elliptic equations: The convergence result, the multiple-scale method, Tartar's method of oscillating test functions, the two-scale convergence method, homogenization in linearized elasticity, homogenization of the heat equation, homogenization of the wave equation, general approaches to the non-periodic case, bounds for effective parameters, and gamma-convergence of integral functionals.

Objectives of the course

Knowledge:
After passing the course, the student is expected to be able to

• Demonstrate knowledge of the main concepts of homogenization theory for partial differential equations and integral functionals.
• Use the basic results of this theory and reach a high level in the forefront of knowledge connected to homogenization of several kinds of second-order boundary value problems.

Skills:
After passing the course, the student is expected to be able to

• Formulate relevant research problems connected to homogenization of steady and non-steady heat equations, elasticity problems.
• Solve problems of high international standard devoted to homogenization of second-order boundary value problems.

Habits of mind:
After passing the course, the student is expected to be able to

• Formulate and solve complicated problems by using homogenization techniques in a way that enables communication of research on high level connected to homogenization of second-order boundary value problems.

Language of instruction and examination

Teaching methods

Assessment

Recommended reading/syllabus