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.

19 pages

References in corpus (4)

• On Rank-One Convex Functions that are homogeneous of Degree One
• Rank-one convexity implies polyconvexity for isotropic, objective and isochoric elastic energies in the two-dimensional case
• Energy minimisers with prescribed Jacobian
• Extremal rank-one convex integrands and a conjecture of Šverák