### Homogenization of partial differential equations (Winter Semester 2014/15)

- Lecturer: PD Dr. Andrii Khrabustovskyi
- Classes: Lecture (0108000), Problem class (0108100)
- Weekly hours: 3+1

**Description**

In many problems of physics and mechanics processes in media with rapidly oscillating spatial local characteristics are studied. There are two main types of such media:

- *composite materials* in which the physical processes are described by PDEs with highly oscillating (with respect to spatial variables) coefficients. For example: a medium consisting of a basic material with a lot of small inclusions made of another materials (see left picture);

- *strongly perforated media* in which the physical processes are described by boundary value problems in domains with complicated geometry. For example: a domain with a lot of small holes (see right picture).

It is practically impossible to solve these problems either by analytical or numerical methods. However when the scale of the microstructure of the medium is much smaller than the scale of the physical process under consideration, the medium has homogenized characteristics (which, in general, differs from local ones). The problem of the homogenization theory is to find these characteristics and using them to construct the homogenized model approximating the initial one and giving global description of the physical process in microinhomogeneous media.

The course devoted to some basic problems and methods of the homogenization theory.

**Requirements**

Basic knowledges in functional analysis (Banach and Hilbert spaces, linear operators, weak and strong convergences etc.) and partial differential equations (Sobolev spaces, weak solutions etc.)

**Literature**

**1.** G. Allaire, Shape optimization by the homogenization method, Springer, New York, 2002.

*only Chapter 1.* *Short version:* G. Allaire, Introduction to homogenization theory, EA-EDF-INRIA school on homogenization, 13-16 December 2010. on-line

**2.** V. Marchenko, E. Ya. Khruslov, Homogenization of partial differential equations, Birkhauser, Boston, 2006.

*only Chapters 1, 4.*

Other references will be given through the lectures.

Schedule | |||
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Lecture: | Wednesday 15:45-17:15 (every 2nd week) | K2 Kronenstr. 32 Geb. 1.93 | Begin: 24.10.2014 |

Friday 14:00-15:30 | K2 Kronenstr. 32 Geb. 1.93 | ||

Problem class: | Wednesday 15:45-17:15 (every 2nd week) | K2 Kronenstr. 32 Geb. 1.93 | Begin: 29.10.2014 |

Lecturers | ||
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Lecturer, Problem classes | PD Dr. Andrii Khrabustovskyi | |

Office hours: by appointment | ||

Room 3.037 Kollegiengebäude Mathematik (20.30) | ||

Email: andrii.khrabustovskyi@kit.edu |