Johannes Zimmer, Mechanical Engineering, CalTech
"Nonconvex Symmetric Functions in Continuum Mechanics"
Applications in continuum mechanics often require the energy function to be nonconvex. In the calculus of variations, different notions of convexity play an important role. In both cases, few examples of specific functions are known (since, usually, physical laws impose additional symmetry conditions). The aim of this talk is to present some methods to construct such functions in a systematic way.
First, I will discuss nonconvex functions which are invariant under a discrete symmetry group and depend on the temperature as a parameter. These functions may serve as energy functions for phase transitions in crystals. This approach yields a formal description of all $C^\infty$ potentials.
In the second part, polyconvex frame-indifferent functions are considered. The aim is to extend Ball's theorem (ARMA '77) on isotropic polyconvex functions. I present a sufficient condition describing a large class of frame-indifferent polyconvex functions.
THURSDAY, December 6, 2001