15. ÖMG-Kongress
Jahrestagung der Deutschen Mathematikervereinigung

16. bis 22. September 2001 in Wien

Dienstag, 18. September 2001, 10.00-11.00, Audimax der Universität Wien


Wigner transforms and homogenization limits

Norbert Julius Mauser, Universität Wien


In the last decade the method of Wigner transforms has proven to be very succesful in the context of (semi)classical limits and general homogenization limits [1]. The basic idea is to transform a PDE in ``physical'' space into sort of a kinetic equation in ``phase-space''.

For problems with a distinguished scale of oscillations Wigner measures are clearly more convenient than e.g. H-measures.

We present recent results on the adoption of Wigner transforms to homogenization of periodic structures, the so called Wigner series. The outstanding problem of the semiclassical limit of the nonlinear Schrödinger-Poisson problem [2] in a crystal has recently been solved by such Wigner Bloch measures [3], thus giving the first rigorous justification of the ``semiclassical equations'' of solid state physics

[1] ``Wigner transforms and Homogenization limits'', P. Gérard, P. A. Markowich, N.J. Mauser and F. Poupaud, Comm.Pure and Appl.Math., 50 (1997) 321-377
[2] ``Sur les Mesures de Wigner'', P.L. Lions and T. Paul, Revista Mat. Iberoamericana, 9 (1993) 553-618
[3] ``Semiclassical limit for the Schrödinger-Poisson equation in a crystal'', P. Bechouche, N.J. Mauser and F. Poupaud, Comm.Pure and Appl.Math., XXX (2001) 1-42

E-Mail: norbert.mauser@univie.ac.at

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