Study of polyconvexity in some problems in the calculus of variations

Abstract

This study seeks to investigate the concept of symmetric polyconvex functions in higher-dimensional spaces. By advancing the methodology introduced by Boussaid et al. for two-dimensional and three-dimensional cases, we introduce an innovative characterization of symmetric polyconvex functions in higher dimensions. Our principal fiding reveals that the requisite condition for symmetric polyconvexity of a function f is its ability to be formulated as a convex function that incorporates the matrix and its second-order minors, exhibiting a non-increasing tendency in a specifi sense with respect to the second-order minor variable. Additionally, we propose and scrutinize the concept of S-positive semi-defiite matrices, which is crucial to our characterization. This new characterization also enables the identifiation of the class of symmetric polyconvex quadratic forms and demonstrates the absence of non-trivial symmetric poly-affi functions.