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Monday, 26 August 2019
Time Speaker Title Resources
09:45 to 10:45 A. K. Nandakumaran General Introduction to Homogenization

Multi scales arise in many physical and industrial problems. Many industrial constructions include very complicated structures. Homogenization is a branch of science where we try to understand microscopic structures via a macroscopic medium. This study is basically developed from material science in the creation of composite materials though the present application is much far and wide. It has applications in {\it composite media, porous domains, laminar structures, domains with rapidly oscillating boundaries}, to name a few. The PDE problems posed on such complicated domains lead to a type of asymptotic analysis known as homogenization.  It is a process of understanding the microscopic behavior of an in-homogeneous medium via a homogenized medium.
There are various methods developed in the last 50 years to understand the mathematical homogenization theory and some them are; Asymptotic Expansion, Energy Method, Compensated Compactness, Two-scale and multi-scale convergence, Gamma Convergence, Bloch Wave Analysis, Method of Unfolding etc. We give a quick introduction on the topic and briefly discuss the aim of the discussion meeting and topics to be discussed in the next two weeks.

10:45 to 11:45 Patrizia Donato Introduction to Sobolev Spaces and Weak Solutions of PDEs

The aim of this welcome lecture is to give a general overview of variational problems, in order to prepare the audience to the following talks. A short introduction to variational PdE’s will be followed by the definition of a class  of Sobolev spaces and its properties. then we introduce the variational formulation and give some existence and uniqueness result for the model equation in the divergence form.  The case of homogeneus Dirichlet boundary conditions and periodic boundary conditions Dirichlet are treated. This last case allows to introduce the corrector  functions appearing in homogenization.

11:45 to 12:15 Break Tea/coffee
12:15 to 13:15 Editha Jose Multiscale Expansion Method for Periodic Homogenization

Homogenization is a way of seeking the average properties of a material out of its components. In this lecture, we focus on the homogenization of periodic structures by considering an asymptotic analysis of the PDEs describing a certain property of a material. The limiting process is done by taking the period to approach zero and obtain a homogeneous model with homogenized coefficients that depend on the coefficients of the components. In this particular lecture, we discuss the multiscale expansion method which is a heuristic one but allows one to formally homogenize a great variety of equations posed in a periodic domain.

13:15 to 14:30 Break Lunch
14:30 to 16:00 Editha Jose Multiscale Expansion Method for Periodic Homogenization

Homogenization is a way of seeking the average properties of a material out of its components. In this lecture, we focus on the homogenization of periodic structures by considering an asymptotic analysis of the PDEs describing a certain property of a material. The limiting process is done by taking the period to approach zero and obtain a homogeneous model with homogenized coefficients that depend on the coefficients of the components. In this particular lecture, we discuss the multiscale expansion method which is a heuristic one but allows one to formally homogenize a great variety of equations posed in a periodic domain.

16:00 to 16:30 Break Tea/coffee
16:30 to 18:00 Tutorials/Discussion Tutorials/Discussion/Lecture
Tuesday, 27 August 2019
Time Speaker Title Resources
09:30 to 11:00 Patrizia Donato The Tartar's method of oscillating test functions and correctors

In this lecture we introduce the Tartar’s method of oscillating test function. Then, using this method, we first prove the classical homogenisation result, then we state and prove the corresponding corrector result.

11:00 to 11:30 Break Tea/coffee
11:30 to 13:00 Ravi Prakash Multi-Scale Convergence
13:00 to 14:30 Break Lunch
14:30 to 15:45 Patrizia Donato Tartar's method and correctors in perforated domains

In this lecture we make use of  the Tartar’s method of oscillating test function adapted to the case of problems posed in domains  periodically perforated by holes of the same size as the period.
We treat first the  case of  homogeneus Dirichlet conditions on the boundary of holes. Then we introduce general variational problems with Dirichlet-Neumann conditions and apply to discuss the homogenisation in the case of the  case of  homogeneus Dirichlet conditions on the exterior boundary and neumann conditions on   boundary of holes. Extension operator are discussed, in order to state the results and show how  the Tartar’s method apply in this case. Finally we state a corrector result.

15:45 to 16:15 Break Tea/coffee
16:15 to 18:00 Tutorials/Discussion Tutorials/Discussion/Lecture
Wednesday, 28 August 2019
Time Speaker Title Resources
09:30 to 11:00 M. Vanninathan H-measure and Applications

Following Tartar, we introduce H-measure for a sequence of functions whose weak limit is zero. We mention some of its properties,particularly, in connection with Compensated Compactness. Next, we discuss its application in obtaining optimal estimates of conductivities of mixtures obtained by mixing two isotropic homogeneous heat conductors via  arbitrary microstructures.

11:00 to 11:30 Break Tea/coffee
11:30 to 13:00 Daniel Onofrei Unfolding Method and Homogenization
13:00 to 14:30 Break Lunch
14:30 to 15:45 Ravi Prakash Multi-Scale Convergence
15:45 to 16:15 Break Tea/coffee
16:15 to 18:00 Tutorials/Discussion Tutorials/Discussion/Lecture
Thursday, 29 August 2019
Time Speaker Title Resources
09:30 to 11:00 M. Vanninathan H-measure and Applications

Following Tartar, we introduce H-measure for a sequence of functions whose weak limit is zero. We mention some of its properties,particularly, in connection with Compensated Compactness. Next, we discuss its application in obtaining optimal estimates of conductivities of mixtures obtained by mixing two isotropic homogeneous heat conductors via  arbitrary microstructures.

11:00 to 11:30 B Tea/coffee
11:30 to 13:00 Patrizia Donato The periodic unfolding method in perforated domains

In this lecture we show how the periodic unfolding method, already introduced in previous lectures, can be extended to the case of perforated domains and describe the corresponding results.

13:00 to 14:30 Break Lunch
14:30 to 15:45 M. Vanninathan H-measure and Applications

Following Tartar, we introduce H-measure for a sequence of functions whose weak limit is zero. We mention some of its properties,particularly, in connection with Compensated Compactness. Next, we discuss its application in obtaining optimal estimates of conductivities of mixtures obtained by mixing two isotropic homogeneous heat conductors via  arbitrary microstructures.

15:45 to 16:15 Break Tea/coffee
16:15 to 17:30 Hari Shankar Mahato Biological Applications
Friday, 30 August 2019
Time Speaker Title Resources
09:30 to 11:00 Daniel Onofrei Unfolding Method and Homogenization
11:00 to 11:30 Break Tea/coffee
11:30 to 13:00 Ravi Prakash Multi-Scale Convergence
13:00 to 14:30 Break Lunch
14:30 to 15:45 Editha Jose TBA
15:45 to 16:15 Break Tea/coffee
16:15 to 17:30 Hari Shankar Mahato Biological Applications
Monday, 02 September 2019
Time Speaker Title Resources
09:30 to 11:00 Daniel Onofrei Unfolding Method and Homogenization
11:00 to 11:30 Break Tea/coffee
11:30 to 13:00 Antonio Gaudiello Oscillating Boundary problems
13:00 to 14:30 Break Lunch
14:30 to 15:45 T. Muthukumar Bloch Wave Analysis
15:45 to 16:15 Break Tea/coffee
16:15 to 18:00 Tutorials/Discussion Tutorials/Discussion/Lecture
Tuesday, 03 September 2019
Time Speaker Title Resources
09:30 to 11:00 Antonio Gaudiello Oscillating Boundary problems
11:00 to 11:30 Break Tea/coffee
11:30 to 13:00 A. K. Nandakumaran Gamma Convergence

The $\Gamma$ convergence was introduced in the seventies by DeGiorgi and Spagnolo to study the convergence of the minimizers and minimal values of a sequence of functionals. These notions can be introduced in the very general setting of abstract topological spaces. But it is more interesting if the functionals are given as integral functionals and it is worthwhile to know whether the $\Gamma$ limit functional is also an integral. This is important in the study of PDE, where quite often the solution of PDE is the minimizer of an associated functional. Here, we study the $\Gamma$ convergence in the context of homogenization problems. In the two lectures, we introduce the general concept of the convergence and in another lecture, Harsha will present its application to homogenization.

13:00 to 14:30 Break Lunch
14:30 to 15:45 T. Muthukumar Bloch Wave Analysis
15:45 to 16:15 Break Tea/coffee
16:15 to 17:30 A. K. Nandakumaran Gamma Convergence
The $\Gamma$ convergence was introduced in the seventies by DeGiorgi and Spagnolo to study the convergence of the minimizers and minimal values of a sequence of functionals. These notions can be introduced in the very general setting of abstract topological spaces. But it is more interesting if the functionals are given as integral functionals and it is worthwhile to know whether the $\Gamma$ limit functional is also an integral. This is important in the study of PDE, where quite often the solution of PDE is the minimizer of an associated functional. Here, we study the $\Gamma$ convergence in the context of homogenization problems. In the two lectures, we introduce the general concept of the convergence and in another lecture, Harsha will present its application to homogenization.
Wednesday, 04 September 2019
Time Speaker Title Resources
09:30 to 11:00 Antonio Gaudiello Oscillating Boundary problems
11:00 to 11:15 Break Tea/coffee
11:15 to 12:45 T. Muthukumar Bloch Wave Analysis
12:45 to 13:45 Break Lunch
13:45 to 14:45 Harsha Hutridurga Gamma Convergence and Homogenization
14:45 to 15:00 Break Tea/coffee
16:15 to 17:30 Patrizia Donato Reserach lecture

In this research I will present some recent homogenization results for several kind of domains.

Thursday, 05 September 2019
Time Speaker Title Resources
09:30 to 10:30 Antonio Gaudiello Oscillating Boundary problems
10:30 to 11:30 Daniel Onofrei Unfolding Method and Homogenization
11:30 to 12:00 Break Tea/coffee
12:00 to 13:00 T. Muthukumar Research lecture
13:00 to 14:30 Break Lunch
14:30 to 15:30 Editha Jose Research lecture
15:30 to 16:00 Break Tea/coffee
16:00 to 17:00 Harsha Hutridurga Research lecture : Homogenization of Reaction-Diffusion systems and related topics.

In this seminar, we address certain periodic homogenization problems of reaction-diffusion systems arising in the context of reversible chemistry. We try to highlight the main difficulties involved in the mathematical analysis of such models. The importance of the entropy dissipation structure of the rate functions involved in reversible chemical kinetics will be emphasised. We will be working with multiplier based energy method as applicable to these systems and the parabolic duality estimates. We also hope to highlight some of the open questions related to these models. For homogenization purposes, we will be employing the method of two-scale convergence, which will be briefly introduced during the seminar.

Friday, 06 September 2019
Time Speaker Title Resources
09:00 to 09:20 Federica Raimondi Research presentation
09:25 to 09:45 Adrien Ceccaldi Research presentation
09:50 to 10:10 Eleanor Gemida Research presentation
10:15 to 10:35 Ivy Carol Belaro Lomerio Research presentation
10:45 to 11:15 Break Tea/coffee
11:15 to 11:35 Abu Sufian Research presentation
11:40 to 12:00 Aiyappan Srinivasan Research presentation
12:05 to 12:25 Rheadel Fulgencio Research presentation
12:30 to 12:50 Vivek Tewary Research presentation
12:50 to 13:00 Conclusion Conclusion