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Towards Characterization of All 3 × 3 Extremal Quasiconvex Quadratic Forms
Author(s) -
Harutyunyan Davit,
Milton Graeme Walter
Publication year - 2017
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21699
Subject(s) - quasiconvex function , mathematics , tensor (intrinsic definition) , quadratic form (statistics) , zero (linguistics) , rank (graph theory) , pure mathematics , quadratic equation , polynomial , isotropic quadratic form , binary quadratic form , square (algebra) , mathematical analysis , combinatorics , regular polygon , quadratic function , convex optimization , geometry , convex analysis , linguistics , philosophy
Given a d × d quasiconvex quadratic form, d ≥ 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 extremal or polyconvex. Also, if the determinant of the acoustic tensor of the form is a perfect square, then the form is either extremal or polycovex, or is a sum of a rank‐1 form and an extremal whose acoustic tensor determinant is identically zero. Here we use the notion of extremality introduced by Milton in 1990. © 2017 Wiley Periodicals, Inc.