Lecture note 45/2020
An introduction to generalized Young measures
Jan Kristensen and Bogdan Raita
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Submission date: 02. Nov. 2020
MSC-Numbers: 49J45, 35G05
Keywords and phrases: Generalized Young measures, lower semi-continuity, a-quasiconvexity, Constant rank, Linear partial differential operators, Weak sequential continuity, Oscillation, concentration
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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, 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 generalized Young measures, 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 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.