Bogdan Raita: An introduction to generalized Young measures

  • Lecturer: Bogdan Raita

Abstract

In the study of PDEs, one cannot realistically hope for better compactness than in the weak topologies, which reasonably describe measurements of physical quantities. The phenomena that are less understood do not usually lead to linear equations, making the interaction between nonlinear quantities and weakly convergent sequences an ubiquitous theme in the study of nonlinear PDEs.

In this course, we will learn about generalized Young measures (GYM), which are especially useful tools to describe the effective limits of nonlinearities applied to weakly convergent sequences in Lebesgue spaces. In particular, these objects efficiently keep track of concentration and oscillation effects, which are the main obstructions to strong convergence. Thus, Young measures are naturally used in many branches of PDE theory, of which we will focus on their role in the Calculus of Variations, where the notion first emerged from the ideas of L.C. Young.

We will thoroughly explore the functional analytic and measure theoretic realities of GYM, with a focus on identifying oscillation and concentration effects, or lack thereof. We will use this basic understanding to study weak sequential continuity and lower semi-continuity of energy functionals

$$

E[v]=\int_{\Omega} F(v(x))\mathrm{d\,}x

$$

on linear PDE constrained subsets of Lebesgue spaces. Such problems arise in elasticity, plasticity, composites, fluids, and electromagnetism, to name a few; more abstract applications arise in differential geometry and geometric measure theory. Time permitting, we will present the main result of my recent preprint arXiv:1912.09190, joint with Jan Kristensen.

Contents

About the Autor

Prof. Dr. Eberhard Zeidler wurde 1940 in Leipzig geboren. Dort studierte er Mathematik und Physik. 1974 wurde er zum ordentlichen Professor für Analysis an die Universität Leipzig berufen. Zusammen mit Prof. Dr. Jürgen Jost und Prof. Dr. Stefan Müller gründete er 1996 das Max-Planck-Institut für Mathematik in den Naturwissenschaften in Leipzig und war von 1996 bis 2003 dessen geschäftsführender Direktor. Er ist Mitglied der Deutschen Akademie der Naturforscher Leopoldina. Für sein Lebenswerk erhielt er den Alfried Krupp Wissenschaftspreis 2006 der Alfried Krupp von Bohlen und Halbach-Stiftung.
Prof. Zeidler starb im November 2016.

Figure 1: Einstein's Cross

27.03.2020, 02:30