Prerequisites

Real Analysis, Functional Analysis, Measure Theory; specifically, knowledge of:

• Banach and Hilbert spaces, dual spaces, and convergence in these spaces

• Lebesgue integral and Lp-spaces

• Convergence criteria for Lp functions: Monotone convergence theorem, dominated convergence theorem, Fatou's lemma

• Fundamental properties of weak convergence and weak compactness in Lp spaces

The necessary background on these topics can be found in Chapters 2-5 in the book by H. W. Alt, or in Chapter 4 of H. Brezis for Lp-spaces.

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 introduce 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:

• 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

• Deduce Euler-Lagrange equation (weak and strong form) via the first variation

• Establish fundamental properties of solutions of minimization problems (e.g., uniqueness, regularity)

• Use relaxation theory to characterize the limiting behavior of minimizing sequences

Rules of Homework and Exam

The course counts for 8 EC.

Homework is bi-weekly.

The exam will be written if sufficiently many students are taking it.
Otherwise, it will be an oral exam.

The students pass the course if the final grade is at least 6.
If the exam grade is at least 5 and if the homework grade (obtained by solving the problem sheets) is better than the exam grade, the final grade will be the weighted average of both:

75% exam   /   25% homework grade.

Rounding will be done at the end of the above computation.

Lecture notes/literature

Lecture notes will be developed and made available during the course.
They will contain all the material covered in class, and also additional material (not necessary for the exam) to give a more comprehensive overview of the subject.
There is no need to buy any book.

For further reading we recommend:

For the background material on functional analysis and PDEs:

• H. W. Alt, Linear functional analysis: An application-oriented introduction. Springer, 2016

• H. Brezis, Functional Analysis, Springer, 2011

• L.C. Evans, Partial differential equations. American Mathematical Society, 1998

For the main content of the course:

• Braides, Γ-convergence for beginners. Oxford Lecture Series in Mathematics and Its Applications, Oxford University Press, 2002. SIAM, Philadelphia, 2014

• B. Dacorogna, Introduction to the calculus of variations. Imperial College Press, 2015

• F. Rindler, Calculus of Variations. Universitext, Springer, 2018