Towards characterization of all $3\times3$ extremal quasiconvex quadratic forms

TL;DR

This paper characterizes all 3x3 extremal quasiconvex quadratic forms by analyzing the properties of their acoustic tensor determinants, establishing conditions under which these forms are extremal, polyconvex, or sums involving rank-one forms.

Contribution

It extends the understanding of extremal quasiconvex quadratic forms by linking their extremality to the algebraic properties of their acoustic tensor determinants, especially in the 3x3 case.

Findings

For d≥3, extremality is linked to the acoustic tensor determinant being an irreducible extremal polynomial.

In the 3x3 case, if the determinant is not a perfect square, the form is extremal.

When the determinant is zero or a perfect square in 3x3 forms, the form is either extremal, polyconvex, or a sum involving a rank-one form.

Abstract

Given a $d\times d$ quasiconvex quadratic form, $d\geq 3,$ we prove that if the determinant of its acoustic tensor is an irreducible extremal polynomial that is not identically zero, then the form itself is an extremal quasiconvex quadratic form, i.e. it loses its quasiconvexity whenever a convex quadratic form is subtracted from it. In the special case $d=3,$ we slightly weaken the condition, namely we prove, that if the determinant of the acoustic tensor of the quadratic form is an extremal polynomial that is not a perfect square, then the form itself is an extremal quadratic form. In the case $d=3$ we also prove, that if the determinant of the acoustic tensor of the form is identically zero, then the form is either an extremal or polyconvex. Also, if the determinant of the acoustic tensor of the form is a perfect square, then the form is either extremal, polycovex, or is a sum of a rank-one form and an extremal, whose acoustic tensor determinant is identically zero. Here we use the notion of extremality introduced by Milton in [\ref{bib:Mil3}]