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

• Lecturer:
• Classes: Lecture (1054), Problem class (1055)
• Weekly hours: 2+1

## Excercise class begins: Tuesday, October 27th

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: Please get in contact by email.
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

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.

# References

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