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)|
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.
- 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.
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.