Real Analysis, Functional Analysis, Measure Theory, in particular, knowledge of:
- Banach and Hilbert spaces, dual spaces, and convergence in these spaces
- Linear operators: definition, basic properties and compact operators
- Lebesgue integral and Lp-spaces
- Convergence criteria for Lp functions: Monotone convergence theorem, dominated convergence theorem, Fatou's lemma
The necessary background on these topics can be found in Chapters 2-5 in the book by Hans Wilhelm Alt, Linear Functional Analysis: An Application-Oriented Introduction, Springer.
Master-level courses on partial differential equations and functional analysis will be helpful but are not mandatory.
Aim of the course
The calculus of variations is an active area of research with important applications in science and technology, e.g. in physics, materials science or image processing. Moreover, variational methods play an important role in many other disciplines of mathematics such as the theory of differential equations, optimization, geometry, and probability theory.
The goal of this course is to give an introduction to different facets of this interesting field, which is concerned with the minimization (or maximization) of functionals.
By the end of the course, the student should be able to
- derive variational models to describe real world situations
- deduce Euler-Lagrange equations via the first variation
- exploit methods from the theory of differential equations to identify explicit representations or other properties (e.g. regularity) of solutions to variational problems
- apply the direct method in the calculus of variations to prove existence of minimizers
- deduce lower semicontinuity of integral functionals based on convexity properties of the integrands
- identify non-existence of solutions to variational problems and use relaxation theory to characterize behavior of almost minimizers
- characterize the asymptotic behavior of parameter-dependent variational problems via Γ-convergence
- Dr. Stefanie Sonner (RU), email: firstname.lastname@example.org (week 6-13)
- Dr. Oliver Tse (TU/e), email: email@example.com (week 15-22)
- Gabriele Fissore (RU), email: firstname.lastname@example.org (teaching assistant)