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Variational methods and applications to PDEs (Winter Semester 2009/10)

Note --- new room: S33 (old math building)

Lecture begins: Monday, October 19th

Excercise class begins: Tuesday, October 27th

Lecture: Monday 14:00-15:30 S 33 (old math building)
Problem class: Tuesday 15:45-17:15 S 33 (old math building)
Lecturer, Problem classes Prof. Dr. Wolfgang Reichel
Office hours: Monday, 11:30-13:00 Before you e-mail: call or come!
Room 3.035 Kollegiengebäude Mathematik (20.30)
Email: Wolfgang.Reichel@kit.edu
Lecturer, Problem classes Prof. Dr. Michael Plum
Office hours: Please get in contact by email.
Room 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.

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