MIGSAA Variational Methods for PDEs (Semester 2, 2019)

Weeklies

Week 1: John gives an intensive course on Sobolev spaces and the calculus of variations in one dimension. Notes here.

Week 2: We discussed the functional analytic framework behind the calculus of variations, following the first few pages of the book by Struwe on Variational Methods.

Week 3: We discussed weak lower semicontinuity for functionals given by integrals and its relation to the convexity of the integrand, following Struwe's notes on Variational Methods. In the second part of the lecture we explained weak lower semicontinuity for polyconvex integrands.

Week 4: We discussed compensated compactness following Section 3 in Struwe's notes and prove weak lower semicontinuity for polyconvex integrands.

Week 5: We discussed the basic theory of finite element methods for linear problems (coercive bilinear forms, Cea's Lemma). Then we discussed the discretization of convex nonlinear problems and Gamma-convergence, following Sections 4.1.1 and 4.1.2 in Bartels' notes.

Week 6: We continue with the discretization of nonlinear problems, following Chapter 4 in Bartels' notes.

Week 7: We finish the discussion of discretizations in Bartels' notes and introduce the functional analytic set-up for gradient flows.

Week 8: We discuss the existence of solutions to general gradient flows.

Week 9: We finish the discussion of gradient flows and begin with a discussion of the non-uniqueness of weak solutions to nonlinear PDEs a la Nash and de Lellis-Szekelyhidi.

Week 10: We discuss the convex integration scheme of Nash and its recent application to problems in fluid mechanics.