Day 3 Titles and Abstracts



Bounds on dispersion tensor in periodic media

Loredana Smaranda
University of Pitesti, Romania
smaranda -at- dim.uchile.cl

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In this talk, we consider a periodic media and we study the dependence of the dispersion tensor or Burnett coefficients in terms of the microstructure We treat one-dimensional and laminated structures, and also wegive some perspectives on other cases in higher dimension.

In low contrast periodic media, interesting properties of the sign of this tensor are found. Surprisingly, these depend on the microstructure only through the local proportion parameter and in some cases, they do not depend on the microstructure at all.

Considering the general one dimensional periodic medium, we completely describe the set in which the dispersion coefficient lies, as the microstructure varies preserving the volume proportion. In higher dimension, we study properties on the dispersion tensor for laminated and Hashin structures and we characterize the bounds of this tensor in terms of some geometric properties on the reference cell.

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Non-classical spectral boundary homogenization problems

M. Eugenia Pérez
Universidad de Cantabria, Spain
maria.perez -at- unican.es

Slides not available

Spectral homogenization problems for elliptic operators in bounded domains with strongly oscillating boundary conditions have been approached in the literature for a long time. A common fact to many of these problems is that there is convergence of the spectrum towards the spectrum of the homogenized problem when the parameter which measures the periodicity of the structure converge towards zero. This convergence holds with conservation of the multiplicity, and there is a certain convergence for the eigenfunctions.

When the physical characteristics of the medium are also strongly perturbed near the boundary or the spectral appears in the strongly oscillating boundary conditions , obtaining convergence for the fist eigenvalue or the first eigenfunction can be open problems.

In this talk we provide an overview on these spectral problems and on the techniques developed recently to give some information on the structure of the vibrations of the models in which they arise (cf., for instance, the vibrations of blocks of elastic materials with rough walls, riveted panels along lines, flaws arising in the fabrications of materials, slip-weakening models for periodic systems of faults, etc.).

Among other things, let us mention that constructing the so-called quasimodes allows describing low and high frequency vibrations as has been recently highlighted in several papers . On the other hand, a reformulation of the spectral problems introducing new spectral parameters and spaces different from those where the problems are naturally posed allow obtaining information on the asymptotics for the low frequencies and the associated eigenfunctions.

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On stability estimates for backward heat conduction problem

Thamban Nair
Indian Institute of Technology, Chennai, India
mtnair -at- iitm.ac.in

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In applications, one would like to determine the temperature profile $ u(x,t)$ of a heat conducting body from the knowledge of the temperature at a particular time $ t=T$ for $ T>0$. This problem is a combination of well-posed and ill-posed problems: The problem of determination of temperature at time $ t>T$ is well-posed, whereas the problem of determination of the temperature at time $ t<T$ is ill-posed.

For the ill-posed problem, the so called 'backward heat conduction problem (BHCP)', it is important to know the stability estimates based on certain a priori source conditions on the unknown quantity, and also to have a regularization procedure which provides stable approximate solutions to the problem.

In this talk, we shall discuss the above issues. We shall see that the stability issues for the BHCP of determining the initial temperature $ u(x,0)$ is quite different from that for $ u(x,t)$ with $ 0<t<T$, and require more sofisticated anlalytic tools, which have been developed in recent years.

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On cutting cloth, according to Chebyshev

Etienne Ghys
Ecole Normale Superiore, Lyon, France

Slides not avilable

On August 28th, 1878, Chebyshev gave a talk with the same title in Paris. Given a shape like an elbow for example, how should we cut a piece of cloth to cover it, avoiding folds...Chebyshev gave several concrete examples. I would like to revisit this kind of questions and I'll discuss in particular a good way to cloth a sphere or a large domain in this Poincare disc.

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Pullback Equation: A Survey

Saugata Bandyopadhyay
Indian Institute of Science Education & Research, Kolkatta
saugata.bandyopadhyay -at- iiserkol.ac.in

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Pullback equation is a non-linear system of partial differential equations involving differential forms. In this talk, we will discuss recent developments on issues of existence and regularity of solutions of the pullback equation.

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