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

Lecturer: | Prof. Dr. Wolfgang Reichel , Prof. Dr. Michael Plum |
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Classes: | Lecture (1054), Problem class (1055) |

Weekly hours: | 2+1 |

Lecture: | Monday 14:00-15:30 | S 33 (old math building) |
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Problem class: | Tuesday 15:45-17:15 | S 33 (old math building) |

Lecturer, Problem classes | Prof. Dr. Wolfgang Reichel |
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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: Tue 13:15 - 14:15 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:

- 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.

# 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