Automatic quasiconvexity of homogeneous isotropic rank-one convex integrands

TL;DR

This paper investigates the quasiconvexity and polyconvexity of certain homogeneous, isotropic, rank-one convex integrands, establishing conditions under which these integrands are quasiconvex or polyconvex, with implications for calculus of variations.

Contribution

It provides new conditions for quasiconvexity and polyconvexity of rank-one convex integrands based on homogeneity and isotropy, extending understanding in nonlinear elasticity.

Findings

Integrands are quasiconvex at conformal matrices if p ≤ 2 and Burkholder integrand is quasiconvex.

Positive part of Burkholder integrand is polyconvex for p ≥ 2.

Integrands are polyconvex at conformal matrices for p ≥ 2.

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 $p \leq n = 2$ we prove, conditional on the quasiconvexity of the Burkholder integrand, that the integrands in this class are quasiconvex at conformal matrices. If $p \geq n = 2$, we show that the positive part of the Burkholder integrand is polyconvex. In general, for $p \geq n$, we prove that the integrands in the above class are polyconvex at conformal matrices. Several examples imply that our results are all nearly optimal.