The Edmund Landau Minerva Center for Research in Mathematical Analysis and Related Areas

THE LANDAU LECTURES 2008/2009

Professor Paul Rabinowitz (University of Wisconsin-Madison)

Thursday, December 18th and 22nd, 2008

Towards an Aubry- Mather Theory for a class of PDE's
In the late 1980's, Jürgen Moser initiated a PDE version of the work of Aubry and of Mather for a class of dynamical systems. Moser was interested in solutions possessing certain minimality and 'monotonicity' properties. Over the past few years, there have been several extensions of this work.
In the first lecture, a survey of this research will be presented. In particular the application of minimization methods to find certain basic solutions of the equations and the use of the basic solutions to construct more complex solutions which undergo multiple transitions will be described.
In the second lecture, we complete our survey, take a closer look at some of the results, and discuss further types of solutions.
Wednesday, December 24th, 2008

Variational gluing
The multi-transition solutions mentioned in lectures 1-2 are generally obtained by what may be called variational gluing arguments. Minima of a functional are in a sense glued together to get new critical points and new solutions of the associated Euler equation. In other settings one deals with mountain pass critical points of a functional rather than minima. In this lecture, we will discuss these matters and also show how to construct hybrid solutions by gluing minimizers and mountain pass solutions.

Last updated: Oct. 28th, 2008