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Introduction to the calculus of variations (SoSe 23)

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Description:

The calculus of variation is, to put it simply, the study of critical points of functionals (functions of functions), which is often used in minimization problems. It connects many domains of mathematics such as PDEs, functional analysis, geometry,... and finds applications in a wide variety of physical problems such as in the determination of geodesics (minimizing distances), minimal surfaces, in classical and quantum mechanics (minimizing energies) etc. In this course, we will discuss both the theory and and the applications to some (and more) of the aforementioned problems.

Content:

(might change during the lecture)
  • Finite dimensional case (minimizing functions on a vector space or on a manifold).
  • The classical method: analyzing the critical points when their exist.
  • Hamiltonian formulation
  • The direct method: proof of existence of critical points. Includes: a crash course on functional analysis and Sobolev spaces.

Audience:

The course will be of interest for Master students of mathematics (WP 37 / WP 42), or motivated Bachelor students, who want to intensify their knowledge in analysis and partial differential equations.

Credits:

6 ECTS.

Schedule:

- Lectures (and exercices): Tuesday 10:15-11:45 and Thursday 10:15-11:45, Room: B045
- Exam on Monday 24.07 from 09:00 to 12:00, Room: B252

References:

  • B. van Brunt, The Calculus of Variations, Universitext, Springer, 2004
  • B. Dacorogna, Introduction to the Calculus of Variations. Imperial College Press, 2004
  • P. Blanchard, E. Brüning, Variational Methods in Mathematical Physics: A unified approach, Springer, 1992
  • L.C. Evans, Partial Differential Equations, AMS, 1998
  • M. Lewin, Describing lack of compactness in Sobolev spaces, unpublished lecture notes "Variational Methods in Quantum Mechanics", 2010, Université de Cergy-Pontoise
  • E. H. Lieb, M. Loss, Analysis, Graduate studies in mathematics, AMS, 2001
  • D. Gilbarg, N.S. Trudinger, Elliptic partial differential equations of second order. Vol. 224. No. 2. Berlin: springer, 1977.
  • V.I. Arnold, Mathematical Methods of Classical Mechanics, Springer, 1989
  • D. Cioranescu, P. Donato, An introduction to Homogeneization, Oxford University Press, 2000

Exercise sheets