Automatic Quasiconvexity of Homogeneous Isotropic Rank-One Convex Integrands

Abstract

We consider the class of non-negative rank-one convex isotropic integrands on \(\mathbb {R}^{n\times n}\) which are also positively p-homogeneous. If \(p\le n=2\) we prove, conditional on the quasiconvexity of the Burkholder integrand, that the integrands in this class are quasiconvex at conformal matrices. If \(p\ge n =2\), we show that the positive part of the Burkholder integrand is polyconvex. In general, for \(p\ge n\), we prove that the integrands in the above class are polyconvex at conformal matrices. Several examples imply that our results are all nearly optimal.