This course is scheduled over 2 x 2 hour-sessions, on Wednesdays 4 and 11 March. Schedule as per below.
The course is devoted to some aspects of asymptotic analysis and homogenization theory of differential operators. We introduce basic notions, give necessary definitions and prove basic theorems. Then we apply this statements and use this approach to main problems in singularly perturbed micro inhomogeneous domains. We give some examples and study the behavior of porous media, perforated and skeleton structures and describe the effective models for these examples.
Lecture 1 (10:00-11:00, Wednesday, March 4, 2015) Some facts of functional analysis (weak derivatives, convergences, weak solutions, Lax-Milgram Theorem, Rellich Theorem).
Lecture 2 (11:00-12:00, Wednesday, March 4, 2015) Compensated compactness ("Curl-div" condition, convergence of solutions)
Lecture 3 (10:00-11:00, Wednesday, March 11, 2015) Problems with rapidly changing type of boundary conditions.
Lecture 4 (11:00-12:00, Wednesday, March 11, 2015) "Strange term" for perforated domains.
Recommended literature: G.A. Chechkin, A. L. Piatnitski, A. S. Shamaev "Homogenization: Methods and Applications" Translations of Mathematical Monographs. V. 234. Providence: AMS, 2007 ISBN: 978-0-8218-3873-0