Variational methods and applications to PDEs (Wintersemester 2009/10)
- Dozent*in: Prof. Dr. Wolfgang Reichel , Prof. Dr. Michael Plum
- Veranstaltungen: Vorlesung (1054), Übung (1055)
- Semesterwochenstunden: 2+1
Note --- new room: S33 (old math building)
Lecture begins: Monday, October 19th
Excercise class begins: Tuesday, October 27th
| Termine | ||
|---|---|---|
| Vorlesung: | Montag 14:00-15:30 | S 33 (old math building) |
| Übung: | Dienstag 15:45-17:15 | S 33 (old math building) |
| Lehrende | ||
|---|---|---|
| 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: Kontakt via E-Mail. | ||
| Zimmer 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:
- weak convergence, lower-semicontinuity, convexity
- first variation, Euler-Lagrange equation, Gateaux- and Fr'echet-differentiability
- Sobolev spaces, weak solutions of elliptic PDEs
- constraint optimisation, Lagrange multipliers
- 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.