Masterseminar: Direct Methods in the Calculus of Variations - Einzelansicht

We will discuss various ways in which the direct method can be used in settings in which its application may not be immediate. Examples are the compensated compactness method exploiting special differential structures of certain problems, and the concentration-compactness principle which allows to deal with problems that are invariant under non-compact group actions, such as translations on Euclidean space. Additionally, we will discuss methods to prove existence of critical points of saddle type, exploring problems in the Calculus of Variations that do not involve global energy minimization. Depending on interest, we may also talk about a number of geometric nonlinear PDEs, such as the Yamabe problem or harmonic maps and their corresponding heat flow.

The seminar will be based on the book „Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems” by Michael Struwe.