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Department of Mathematics

Photo: UHH/MIN/Christina Brandt

Variational Calculus

Summer Term 2024

Instructor: Thomas Schmidt.

Lecture (weekly, first on April 2nd, starting time alternates):

  • Tue, 8-10, H3 or Tue, 9-11, H3/R1240

Exercise class (biweekly (!), first on April 2nd):

  • Tue, 10-12, R1240

Relevance: The course is eligible in the Master Program in Mathematics, Industrial Mathematics, Mathematical Physics, or Economathematics. Further, it is mandatory in the Intermaths Program and generally recommended to everyone with interests in analysis.

ECTS points: The module (lecture plus exercise class) counts 6 ECTS points.

Prerequisites: The lecture builds on basic knowledge in analysis (including the theory of Lebesgue integration) and linear algebra.

Contents: The lecture gives an introduction to the calculus of variations, a classical mathematical discipline, which has its roots in the 19th century but has remained an active research area ever since. It is concerned with minimization problems which depend on a function as the unknown (or, in the geometric branch, rather on a surface or a set) and thus cope with infinitely many degrees of freedom. Existence theorems for (generalized) solutions to such problems can typically be obtained in Sobolev spaces or the space of Lipschitz functions, and the first-order conditions for solutions open up a close and distinctive connection to ODEs and PDEs.

Literature: Some books on the calculus of variations are:

  • G. Buttazzo, M. Giaquinta, S. Hildebrandt, One-Dimensional Variational Problems, Oxford University Press, 1998,
  • B. Dacorogna, Introduction to the Calculus of Variations, Imperial College Press, 2004,
  • B. Dacorogna, Direct Methods in the Calculus of Variations, Second Edition, Springer, 2008,
  • I. Fonseca, G. Leoni, Modern Methods in the Calculus of Variations: Lp Spaces, Springer, 2007,
  • E. Giusti, Direct Methods in the Calculus of Variations, World Scientific, 2003,
  • F. Rindler, Calculus of Variations, Springer, 2018.

In addition, preliminary topics such as functional analysis and Sobolev spaces are treated e.g. in:

  • L.C. Evans, Partial Differential Equations, American Mathematical Society, 1998,
  • L.C. Evans, R.F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, 1992,
  • G. Leoni, A First Course on Sobolev Spaces, Second Edition, 2017,
  • W. Rudin, Functional Analysis, Second Edition, 1991,
  • (in German) D. Werner, Funktionalanalysis, 2007.