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Arbeitsgruppe Nichtlineare Partielle Differentialgleichungen

Kollegiengebäude Mathematik (20.30)
Zimmer 3.029

Karlsruher Institut für Technologie
Institut für Analysis
Englerstraße 2
76131 Karlsruhe

Mo-Fr 10:00-12:00, sowie Di+Do nachmittags

Tel.: 0721 608 42064

Fax.: 0721 608 46530

Variational methods and applications to PDEs (Wintersemester 2009/10)

Dozent: Prof. Dr. Wolfgang Reichel , Prof. Dr. Michael Plum
Veranstaltungen: Vorlesung (1054), Übung (1055)
Semesterwochenstunden: 2+1

Vorlesung: Montag 14:00-15:30 S 33 (old math building)
Übung: Dienstag 15:45-17:15 S 33 (old math building)
Dozent, Übungsleiter Prof. Dr. Wolfgang Reichel
Sprechstunde: Montag, 11:30-13:00 bevor Sie mailen:anrufen/vorbeikommen
Zimmer 3.035 Kollegiengebäude Mathematik (20.30)
Email: Wolfgang.Reichel@kit.edu
Dozent, Übungsleiter Prof. Dr. Michael Plum
Sprechstunde: Do 13:30 - 14:30 und nach Vereinbarung
Zimmer 3.028 Kollegiengebäude Mathematik (20.30)
Email: michael.plum@kit.edu


We will consider functionals defined on Banach-spaces and find conditions, such that these functionals possess minimizers or -- more generally -- critical points. Sometimes such minimizers have physical significance, e.g., they may represent energetically optimal configurations in material science (e.g. soap bubbles, buckling plates or beams, orientation of liquid crystals under a magnetic force). A necessary condition for a minimizer is that it has to satisfy the Euler-Lagrange equation (corresponding to the vanishing of the first derivative of a real valued function at a local minimum or local maximum). Often the Euler-Lagrange equation is a nonlinear elliptic partial differential equation. In this lecture we will focus on applying the calculus of variations as a tool to provide existence of solutions to nonlinear elliptic partial differential equations.


  1. weak convergence, lower-semicontinuity, convexity
  2. first variation, Euler-Lagrange equation, Gateaux- and Fr'echet-differentiability
  3. Sobolev spaces, weak solutions of elliptic PDEs
  4. constraint optimisation, Lagrange multipliers
  5. saddle points, mountain-pass lemma

Wherever possible, we will complement the above topics with examples from elliptic partial differential equations.


Multi-variable calculus, functional analysis. A background in partial differential equations is not necessary, but helpful. The lecture is suitable for students in mathematics, physics and engineering.


Functional Analysis Lebesgue Integral
Divergence Theorem

Problem Sheets

Sheet 1 Sheet 2 Sheet 3
Sheet 4 Sheet 5 Sheet 6
Sheet 7 Sheet 8 Sheet 9
Sheet 10 Sheet 11 Sheet 12
Sheet 13


Giaquinta, Hildebrandt: Calculus of Variations I, Springer 1996
Struwe: Variational Methods, Springer 1998
Willem: Minimax theorems, Birkhäuser, 1997