### Wednesday, May 14

1:45 PM-2:30 PM

*Chair: Irene Fonseca, Carnegie Mellon University*

*Peale C Room*

## IP9

Nonconvexity, Singular Perturbation and the Analysis of Material Microstructure

Material microstructure often forms spontaneously on a variety of scales. This formation is driven by the nonconvexity of the underlying energy which favours fine phase mixtures over homogeneous states. The length scales are typically determined by small higher order (singular) contributions to the energy such as surface energy.

Analysis of the full problem is often unfeasible, while suppression of surface energy contributions destroys the information on length scales. One thus tries to derive simpler models in the limit of vanishing surface energy. Most mathematical approaches (e.g. Gamma-convergence) work best if only one small scale is present. The speaker will present ongoing research with G. Alberti to understand multiple scales through a family of probability measures on micropatterns. He will illustrate the concept by a simple model related to austenite/finely-twinned martensite boundaries and will discuss possible applications to thin magnetics films as well as additional difficulties that arise in that context.

**Stefan Müller**

*Max-Planck Institute for Mathematics in the Sciences, Germany*

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*MMD, 12/17/96*