MaPhySto
Centre for Mathematical Physics and Stochastics
Department of Mathematical Sciences, University of Aarhus

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Funded by The Danish National Research Foundation

Professor Michael Vogelius from Rutgers University, visiting the University of Copenhagen (partially funded by the Danish Research Training Council and MaPhySto) gave a graduate course on applied analysis at the University of Copenhagen and the Technical University of Denmark.

Course Schedule

The course was concentrated during the months of November 2000 and January 2001 with 15 hours of lectures each month (for a total of 30 hours of lectures). In November there were 6 two-hour lectures (2 two-hour lectures per week in week nos. 45, 46 and 47) and one three-hour lecture (in week no. 48). The exact dates and times were decided in accordance with the preferences of the participants. In January there were 3 two-hour lectures during week no. 2, and 3 three-hour lectures (on the 15th, the 18th and the 22nd). Two different (but related) topics were covered. It was possible to follow only the lectures on one topic.

The lectures on the 15th, 18th, and 22nd of January 2001 took place at the Technical University. All other lectures were at the University of Copenhagen.

Contents

November 2000 (first part): Homogenization and effective media theories

The main goal of this part of the course was to find the macroscopic (effective) behaviour of models of continuum mechanics with microstructure. These models involve partial differential equations (PDE's), for instance the equations of elasticity or simplified scalar versions, with highly oscillatory coefficients. The resulting (effective) macroscopic models have no microstructure (in the simplest case they involve constant coefficient PDE's) but they are constructed in such a way that the corresponding solutions are the appropriate (weak) limits of the solutions to the models with microstructure (i.e. such that they reproduce the correct ``average'' fields).

We began by studying models with periodic (or ``locally periodic'') microstructure, but then we proceeded to rather arbitrary microstructures and gave a fairly elementary introduction to the theory of $H$-convergence and compensated compactness (primarily due to Murat and Tartar). A central topic were bounds for effective media (f.ex. bounds for the eigenvalues of the tensors associated with the constant coefficient PDE's). This covered for example the simplest examples of geometry independent bounds (sometimes referred to as Hashin Shtrikman bounds) and their rigorous derivation. In this context the course also briefly described various connections to questions of (generalized) optimal design. The course then returned to periodic microstructures to provide a study of the effective modelling of boundary (and interface) layers.

January 2001 (second part): Inverse problems for elliptic differential operators

One often seeks to find solutions to models of continuum mechanics (governed by partial differential equations) assuming knowledge of the material composition. Looking at it from the opposite direction one may of course view ``nature'' as a perfect solver of these equations, and then use accessible measurements of ``nature's'' solution in order to gain information about the material composition.

A simple, but very fundamental question motivating this course is the following:
``To what extent is the coefficient $a(x)$ in the equation $\mathrm{div}(a(x)\ \mathrm{grad}\ u)=0$ determined from just boundary information about solutions $u$?''
Variants of this question can (and will) be asked for the Maxwell Equations, the Equations of Elasticity, the Schröodinger Equation .... etc. Practical applications are found in medical imaging (impedance computed tomography and other forms of tomography) in detection of flaws in metal components (by a method referred to as the eddy current technique).
Specifically the course was to
(1) establish that a (sufficiently smooth) isotropic coefficient $a(x)$ is uniquely determined by the Cauchy (boundary) data of all possible solutions to $\mathrm{div}\ (a(x) \mathrm{grad}\ u)=0$ (in dimension greater than 2),
(2) show that there is not a similar uniqueness theorem for anisotropic coeffients (and discuss what one may determine),
(3) consider the added complications present in 2 dimensions,
(4) prove that a finite number of Cauchy data (typically one or two) makes it possible to determine coefficients $a(x)$ that correspond to a known background medium with a collection of (unknown) cracks or small imperfections,
(5) discuss numerical algorithms.

Credits

Students at the University of Copenhagen wishing to obtain credit for the course were required to attend both parts, and they were then credited ``2 punkter''.

Prerequisites

The course should be accessible to mathematics, physics as well as advanced engineering students with some elementary knowledge of Partial Differential Equations. The speed as well as the particular emphasis was to some extent adjusted to the interests of the audience.

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