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

TL;DR

This paper proves that homogeneous gradient Young measures supported on certain upper triangular matrices are laminates, advancing understanding of Morrey's conjecture by analyzing measures on specific matrix submanifolds.

Contribution

It establishes that measures supported on particular upper triangular matrices are laminates, providing new insights into Morrey's conjecture and matrix measure theory.

Findings

Homogeneous gradient Young measures on specific matrices are laminates.

Results extend to the 3D nonlinear submanifold with zero determinant.

Supports Morrey's conjecture in the context of these matrix measures.

Abstract

It is shown that every homogeneous gradient Young measure supported on matrices of the form $\begin{pmatrix} a_{1,1} & \cdots & a_{1,n-1} & a_{1,n} \\ 0 & \cdots & 0 & a_{2,n} \end{pmatrix}$ 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 $X_{12}>0$.