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.

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.

Functional Analysis Lebesgue Integral
Divergence Theorem

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Literaturhinweise

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