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Workgroup Nonlinear Partial Differential Equations

Secretariat
Kollegiengebäude Mathematik (20.30)
Room 3.029

Address
Karlsruher Institut für Technologie
Institut für Analysis
Englerstraße 2
76131 Karlsruhe
Germany

Office hours:
Mon-Fri 10-12, Tue+Thu 14-16

Tel.: +49 721 608 42064

Fax.: +49 721 608 46530

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

Lecturer: Prof. Dr. Wolfgang Reichel , Prof. Dr. Michael Plum
Classes: Lecture (1054), Problem class (1055)
Weekly hours: 2+1


Schedule
Lecture: Monday 14:00-15:30 S 33 (old math building)
Problem class: Tuesday 15:45-17:15 S 33 (old math building)
Lecturers
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: Thu 13:30 - 14:30 and by appointment.
Room 3.028 Kollegiengebäude Mathematik (20.30)
Email: michael.plum@kit.edu

Content


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.

Topics:

  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.


Prerequisites:

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.

Handouts

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

References

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