ALEX 2018 Workshop: Abstracts

Characterizations of symmetric polyconvex functions

Omar Boussaid${}^{\mathrm{(}\mathrm{1}\mathrm{)}}$, Carolin Kreisbeck${}^{\mathrm{(}\mathrm{2}\mathrm{)}}$, and Anja Schlömerkemper${}^{\mathrm{(}\mathrm{3}\mathrm{)}}$

(1) Faculty of Exact and Computer Sciences, Hassiba Ben Bouali University (Algeria)

(2) Mathematical Institute, Utrecht University (The Netherlands)

(3) Institute of Mathematics, University of Würzburg (Germany)

The notion of symmetric quasiconvexity plays a key role for energy minimization in the setting of geometrically linear elasticity theory. Due to the complexity of the former, a common approach is to retreat to necessary and sufficient conditions that are easier to handle. Based on [1], I will focus on the sufficient condition of symmetric polyconvexity in this talk. I will present characterizations of symmetric polyconvexity in two and three dimensions and show related results on symmetric polyaffine functions and symmetric polyconvex quadratic forms. In particular, I will present an example of a symmetric rank-one convex quadratic form in 3d that is not symmetric polyconvex. The construction of this example is inspired by the famous work by Serre from 1983 on the classical situation without symmetry. Beyond their theoretical interest, our findings on symmetric polyconvexity may turn out useful for computational relaxation and homogenization.