Existence and homogenization results for singular problems... (Lecture 5) by Patrizia Donato

PROGRAM: MULTI-SCALE ANALYSIS AND THEORY OF HOMOGENIZATION

ORGANIZERS: Patrizia Donato, Editha Jose, Akambadath
Nandakumaran and Daniel Onofrei

DATE: 26 August 2019 to 06 September 2019

VENUE: Madhava Lecture Hall, ICTS, Bangalore

Homogenization is a mathematical procedure to understand the multi-scale analysis of various phenomena modeled by partial differential equations (PDEs). It is a relatively new area and has tremendous applications in various branches of engineering sciences like material science, porous media, the study of vibrations of thin structures, composite materials to name a few. Indeed, homogenization can be viewed as a process of understanding a heterogeneous media (where the heterogeneities are at the microscopic level like in composite materials) by a homogeneous media. Mathematically, homogenization deals with the study of asymptotic analysis of the solutions of PDEs by obtaining the equation satisfied by the limit. This limit equation will characterize the bulk or overall behavior of the material, which does not consist of microscopic heterogeneities and can be solved or computed.

Topics to be covered in the workshop include the following but not limited to:

1. Multi-scale problems in applications

2. Introduction to homogenization

3. Techniques in homogenization

4. Recent trends

Plan and Schedule of the Program:

In this discussion meeting, we go through several examples to understand the homogenization procedure in a general perspective together with applications. We also present various mathematical techniques available and some details about the techniques. A tutorial cum problem-solving session will also be conducted so that beginners can learn the material rigorously. In this way, we can train local and international junior students and researchers by equipping them with the theory and applications of multi-scale analysis and homogenization. Furthermore, the speakers will also discuss some ongoing current research and new problems which can foster mentoring or collaboration between students and experts on the area of analysis of multi-scale phenomena.

In addition to the basic material of homogenization, the speakers will be presenting the recent results in their area of expertise and this will be an opportunity for the youngsters to get into this beautiful area of research. Further, every day, we are planning to have a tutorial cum problem-solving session through which the beginners can learn the material in a better way. To make the tutorials/training sessions more effective, we may form small groups and each group may be asked to do some specific material that they can present on the last day. Each small group may be trained by one of the speakers who are available for the entire workshop,

The schedule mainly consists of 4, one-hour lectures per day and a Tutorial cum Problem-Solving Session of one and a half hours which can be extended according to the requirement of the participants.

0:00:00 Existence and homogenization results for singular problems in domain with oscillating interface (Lecture 5)
0:01:25 The singular model problem
0:02:36 Physical motivations
0:03:24 Some references
0:07:51 What happens if we cannot use this strategy?
0:09:36 Hence, a different strategy is used in
0:09:50 The strategy
0:11:08 Same strategy with additional difficulties for the cases presented here concerning
0:11:27 Different articles in collaboration with
0:12:01 The case of a rough periodic interface
0:12:45 The transmission conditions on the rough interface model an imperfect contact between the two components,
0:13:38 The domain
0:15:49 The problem
0:17:10 Assumptions
0:17:57 Moreover
0:18:56 The functional framework
0:20:00 In the same way we define for the flat limit domain
0:20:38 In the sequel we also use the notations
0:21:28 The variational formulation
0:24:49 Existence, boundedness and uniqueness results
0:25:31 The approximates problems for the existence
0:25:58 Proof of the existence
0:29:50 Homogenization results
0:30:57 Further, denoting
0:31:13 The first case
0:32:53 Its variational formulation is
0:33:24 The second case
0:34:08 The third case
0:36:13 Sketch of the proof of the homogenization
0:37:32 The perforated domain
0:38:42 The Problems: Results appeared in
0:38:58 The case of periodic inclusion with a jump
0:39:17 Thanks for your attention!