A Remark on Polyconvex Functions with Symmetry

For a given polyconvex function W, among all associated convex functions g of minors there exists the largest one; this function inherits all symmetry properties of W. For a given associated (not necessarily the largest) function g, one can still find an associated (possibly not the largest) function with the symmetry of W. This function is constructed by averaging of symmetry conjugated functions over the symmetry group of W using Haar’s measure. It follows that if a symmetric polyconvex function W has class k=0,…,∞ associated function, then the averaging produces a symmetric associated function that is class k as well.