The Tartar's method of oscillating test functions and correctors (Lecture 2) 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.

0:00:10 The notion of trace
0:02:31 The trace theorem
0:04:04 The space H1 per(Y)
0:04:59 Periodic boundary conditions
0:06:27 Its variational formulation is
0:09:06 A particular case
0:10:24 Other boundary conditions
0:11:02 Lecture 1: The Tartar's method of oscillating test functions and correctors
0:12:04 Outline
0:12:20 The periodic model of composite materials
0:12:47 A model periodic elliptic problem
0:13:52 The Tartar's method in a fixed domain
0:15:09 Existence and uniqueness of a weak solution
0:16:00 The limit behavior of A(x/ E)
0:17:51 Theorem
0:18:54 As a consequence of the theorem above
0:20:07 Two facts about weak convergence
0:21:37 Theorem 2
0:22:46 Looking for the limit problem
0:23:35 The case n = 1 (S. Spagnolo 1967)
0:25:27 This gives at the limit
0:26:32 Remarks.
0:27:31 The general answer
0:27:50 whose variational formulation is
0:28:08 Hence, uo is the unique solution of
0:29:38 The Tartar's method of oscillating test functions
0:30:50 Theorem Let B0 be the matrix defined by
0:31:47 Proof. To prove the result it is sufficient to show that
0:34:17 From this relation, by choosing
0:34:36 The following important property is straightforward from the previous theorem:
0:35:02 Proof of the homogenization result
0:37:03 Introduce now the (flux) vector
0:40:42 Setting
0:44:46 Proof of the ellipticity of A0
0:47:06 Convergence of the energy
0:48:56 Correctors
0:51:14 Some interesting properties of the corrector matrix Cl are below: Proposition
0:52:28 To prove (ii), let us now introduce the vector function
0:53:21 The corrector result
0:55:40 A crucial lemma
0:57:40 The first term is nothing else than the energy, so that
0:59:18 We obtain
1:02:03 For the last term, we use again the equation satisfied by
1:03:04 Collecting these limits we have
1:05:46 Proof of the corrector result
1:07:45 Thanks for your attention!

[bijayanray]

Mam would there be phi in place of v in equation (at time 10:10) of variational formulation if not so what does the phi means written on extreme right of the same equation?