Direct Methods in the Calculus of Variations

What it is about

You engage with the fundamentals and modern methods of the calculus of variations. By the end you can understand central terms such as coercivity, sublevelness, relaxation and Γ-convergence and apply them to problems in e.g. mathematical physics and materials science.

What you will be able to do

• Understanding of Lebesgue and Sobolev spaces
• Mastery of weak convergence in Lebesgue spaces
• Application of the direct method (coercivity, sublevelness)
• Knowledge of various convexity notions (quasi-, poly-, rank-1-convexity)
• Understanding of weak continuity of the determinant and the null-Lagrangian functionals
• Understanding of equi-integrability and the Vitali lemma
• Foundations of relaxation (convex hull, total variation)
• Fundamental concepts of Γ-convergence with applications to phase transitions and homogenization

What the module consists of

• LecturesTransmission of the theory through lectures, examples and discussions
• Exercise/TutorialDeepening the content through problems, exercises and presentations

Teaching method

• Presentation and lectureIntroduction of concepts, examples and discussion prompts
• Practice problems with solution hintsDeepening and self-check of learning progress
• Student presentations and group discussionsPromotion of active work and exchange