Winter term 2022/23 Lecture Multidimensional Calculus of Variations

Jointly with Thomas Eiter

Timetable

  • lecture Mon 9h-11h
  • lecture Thu 11h-13h
  • exercise Thu 13h-15h

Prerequisites:

Analysis I–III, Linear Algebra I–II, Functional Analysis (Höhere Analysis I) [desirable, but not necessary: Partial Differential Equations (Höhere Analysis II) ]

Planned Topics (according to module description):

Classical Calculus of Variations: Euler-Lagrange equations, necessary and sufficient conditions for weak and strong local extrema.

Modern Calculus of Variations: Existence of global minimizers using the direct method via weak convergence in Sobolev spaces, Lax–Milgram theorem. Rank-one, quasi, and polyconvexity. Extrema under constraints, eigenvalue characterization. Nonlinear elasticity.

Literature:

Functional analysis: [Alt85]

Introductory material: [EkT76, Dac89, Tro96, Dac04]

  • [Alt85] H. W. Alt. Lineare Funktionalanalysis. Springer-Verlag, Berlin, 1985.
  • [Dac89] B. Dacorogna. Direct Methods in the Calculus of Variations. Springer-Verlag,
  • Berlin, 1989.
  • [Dac04] B. Dacorogna. Introduction to the calculus of variations. Imperial College Press, London, 2004.
  • [EkT76] I. Ekeland and R. Temam. Convex Analysis and Variational Problems. North
  • Holland, 1976.
  • [Tro96] J. L. Troutman. Variational calculus and optimal control. Undergraduate Texts
  • in Mathematics. Springer-Verlag, New York, 1996.