Two-by-two upper triangular matrices and Morrey's conjecture

Abstract

It is shown that every homogeneous gradient Young measure supported on matrices of the form $\left(\begin{array}{cccc}{a}_{1,1}& \cdots & a_{1,n-1}& a_{1,n}\\ 0& \cdots & 0& a_{2,n}\end{array}\right)$ is a laminate. This is used to prove the same result on the 3-dimensional nonlinear submanifold of ${\mathbb{M}}^{2\times 2}$ defined by $\det X=0$ and $X_{12}>0$.

12 pages. Accepted version

References in corpus (1)

• Rank-one convexity implies polyconvexity for isotropic, objective and isochoric elastic energies in the two-dimensional case

Cited by in corpus (3)

• On the Energy Scaling Behaviour of a Singularly Perturbed Tartar Square
• Numerical approaches for investigating quasiconvexity in the context of Morrey's conjecture
• Extremal rank-one convex integrands and a conjecture of Šverák