Calculus of Variations
(Lecture course in Winter Term 2019/20)

Instructors: Thomas Schmidt (lecture), Anton Treinov, Lars Poppe (exercise class).

Lecture (first on October 15th):

• Tue, 8-10, H6 and Wed, 12-14, H5

Exercise classes (first on October 22th):

• Tue, 16-18, Room 415

Relevance: The course is eligible as a part of the master in Mathematics, Mathematical Physics, Industrial Mathematics, or Economathematics and is recommended to everyone with interests in analysis. Clearly, other interested participants are also very welcome.

Modular structure and ECTS points: The lecture can be elected as a 6-ECTS module over the first half of the term or alternatively as a 12-ECTS module over the whole term.

Prerequisites: The lecture builds on basic knowledge in analysis (including the theory of Lebesgue integration) and linear algebra, and some familiarity with either PDEs or advanced analysis is recommended. Brief reminders on functional analysis and Sobolev spaces may be given here and there, but in principle the participants are assumed to have or acquire some basic knowledge on these topics.

Contents: Calculus of variations in Sobolev spaces (key topics: semicontinuity and existence, Euler equation, convex duality, quasiconvexity).

Lecture notes: Last version in PDF.

Literature:

• H. Attouch, G. Buttazzo, G. Michaille, Variational Analysis in Sobolev and BV Spaces: Applications to PDEs and Optimization, Second Edition, MOS-SIAM, 2014,
• G. Buttazzo, M. Giaquinta, S. Hildebrandt, One-Dimensional Variational Problems, Oxford University Press, 1998.
• B. Dacorogna, Direct Methods in the Calculus of Variations, Second Edition, Springer, 2008,
• L.C. Evans, Partial Differential Equations, American Mathematical Society, 1998,
• I. Ekeland, R. Temam, Convex Analysis and Variational Problems, SIAM, 1999,
• M. Giaquinta, L. Martinazzi, An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs, Edizioni della Normale, 2012,
• E. Giusti, Direct Methods in the Calculus of Variations, World Scientific, 2003,
• F. Rindler, Calculus of Variations, Springer, 2018,
• M. Struwe, Variational Methods, Springer, 2008.