Linear differential constraints in the Calculus of Variations

Next lectures
14.10.2024, 10:00 (E1 05 (Leibniz-Saal))

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Many physical problems that are described through partial differential equations can also be seen as variational problems, in which we need to find an energy minimisers. Due to natural considerations, physical quantities often satisfy some differential side-constraints; for instance the velocity field of an incompressible fluid is divergence-free. As a consequence, the study of variational problems under such constraints is rather important.

In this lecture, I will attempt to give an overview over different topics connected to linear differential constraints. In particular, topics treated in this lecture (might) include:

1. a (very soft) look at some algebraic properties of differential operators, Korn-type inequalities;
2. A-quasiconvexity and weak lower-semicontinuity;
3. Young measures;
4. different notions of convexity for functions and for sets;
5. some simple schemes of convex integration connected to A-free problems;
6. A-free measures;
7. regularity theory.

Especially topics 5-7 are quite flexible and the detail in treatment might depend on the interest of the audience.

Keywords
A-quasiconvexity, notions of convexity, lower semicontinuity, differential inclusions, convex integration, A-free measures

Prerequisites
Courses in Analysis up to Functional Analysis, Linear Algebra 1 & 2