The calculus of variations is an active area of current research with important applications in science and technology (e.g. physics, materials science, image processing,...),
as well as interesting connections with other disciplines of mathematics such as the theory of differential equations,optimal control or geometry. We will start the course with a brief introduction into this field,which is about minimization (or maximization) of functionals, i.e. real-valued maps defined on(infinite-dimensional) function spaces.The classical or indirect approach to variational problems has its roots in the 18th century. Itis concerned with deriving necessary and sucient conditions for extremal functions, assumingtheir existence. A famous counterexample established by Weierstrass in 1850, however, made themathematics community at the time realize that solutions to variational problems may not existin general. This fundamental observation marked the birth of a new branch called the modernor direct methods in the calculus of variations, which we will focus on in this seminar course.The modern methods help to find criteria for the existence of minimizers, but also to establishproperties of minimizing sequences if minimizers do not exist. On a technical level, functionalanalytical tools as well as convex analysis play a key role in addressing these issues.Another interesting and more recent direction of study regards sequences of variational problemsand their asymptotic behavior. The appropriate concept for limit processes in this context wasintroduced by De Giorgi and Franzoni in 1975 and is based on a special notion of convergence forfunctionals, called gamma-convergence. Over the last decades, this framework has turned out to be aversatile and powerful tool for tackling problems that arise from multiscale modeling.
After the introductory part, we will go more into depth on selected topics, chosen according tothe specic interests of the participants. Here are some suggestions:
Existence of minimizers and the direct methods
- generalized notions of convexity
- lower semicontinuity of integral functionals
- relaxation of variational problems with non-convex integrands
- ntroduction to BV functions
- variational problems with linear growth
- Young measures and relaxation
Gamma-convergence and its applications
- gamma-convergence of integral functionals
- periodic homogenization via gamma-convergence
- variational theory of phase transitions
- dimension reduction via variational convergence
- discrete to continuum limit passages
- evolutionary gamma-convergence
MaterialSome general literature: H. Attouch, G. Buttazzo and G. Michaille, Variational analysis in Sobolev and BV spaces -Applications to PDEs and optimization. SIAM, Philadelphia, 2014. A. Braides, Gamma-convergence for beginners. Oxford University Press, Oxford, 2002. B. Dacorogna, Direct methods in the calculus of variations. Springer, New York, 2008. G. Dal Maso, An introduction to Gamma-convergence. Birkhauser Boston, 1993. L.C. Evans, Partial differential equations. American Mathematical Society, Providence, RI,1998. I. Fonseca and G. Leoni, Modern methods in the calculus of variations: Lp spaces. Springer,New York, 2007.
These references provide material for background reading and serve as a starting point for a moredetailed literature search once the topics have been assigned.
PrerequisitesBasics of analysis, functional analysis and measure theory are expected. Familiarity with Sobolevspaces, the concept of weak convergence, and partial differential equations will be helpful, but isnot required. A brief crash-course during the introductory lectures will give the necessary background.
FormatThis seminar is aimed at master students with a background and interest in pure and appliedanalysis.
OrganizationInterested students should contact Carolin Kreisbeck by email (email@example.com). Please outlineyour general interest in the seminar, your familiarity with prerequisites, and specic preferencesregarding topics. Students are asked to make an appointment at least two weeks before the scheduledpresentation to discuss and settle relevant issues, e.g. structure of the talk, choice of literature,questions about the content, etc.
EvaluationEvery participant is expected to give two seminar talks of 90 minutes each, possibly more, dependingon the number of students. This includes collecting and studying relevant material beforehand,giving a blackboard presentation, preparing a hand-out, and designing homework exercisesfor the other students. Note that hand-in exercises should be approved beforehand by the seminar organizer.
The speaker is responsible for grading the homework assignments. In case of discussionabout the solutions, the final decision will be with the seminar organizer. Attendance at all seminartalks is mandatory. The final grade for the seminar is the weighted average of the homeworkgrades (20%) and the grade for the presentation (80%). The latter takes also the hand-out andthe design of exercises into account.
Learning goalsAfter completion of the course, the student is able to:
- give an overview of modern variational methods and knows how to use them for solving problems.
- acquire a solid understanding of a specic mathematical topic by reading graduate-level textbooks and scientic papers.
- organize material from the literature in analysis into a well-structured and comprehensible presentation.
- choose appropriate means of communicating mathematics both in written and oral form.
- formulate and correct exercises that keep a balance between relevance and feasibility.
- explain specic topics from the content list of the seminar and put them into perspective.
Evaluation matrix presentation (80%)
(includes hand-out and
design of exercises)homework (20%)give an overview of modernvariational methods and know how to use them for solvingproblemsxxacquire a solid understandingof a specic mathematical topic by reading graduate-leveltextbooks and scientic papersx organize material from the literaturein analysis into a well-structured and comprehensible presentationx choose appropriate means ofcommunicating mathematics bothin written and oral formxxformulate and correct exercisesthat keep a balance between relevance and feasibilityx explain specic topics from thecontent list of the seminar andput them into perspectivexx