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A rank-one convex, nonpolyconvex isotropic function on $\textrm {GL}^{\!+}(2)$ with compact connected sublevel sets

Part of: Elastic materials Real functions Functions of several variables

Published online by Cambridge University Press:  02 February 2022

Jendrik Voss ,
Ionel-Dumitrel Ghiba ,
Robert J. Martin  and
Patrizio Neff
Jendrik Voss
Affiliation:
Chair for Nonlinear Analysis and Modeling, Faculty of Mathematics, University of Duisburg-Essen, Thea-Leymann-Str. 9, 45127, Essen, Germany (max.voss@uni-due.de.)
Ionel-Dumitrel Ghiba
Affiliation:
Department of Mathematics, Alexandru Ioan Cuza University of Iaşi, Blvd. Carol I, no. 11, 700506, Iaşi, Romania, Octav Mayer Institute of Mathematics of the Romanian Academy, Iaşi Branch, 700505, Iaşi, Romania
Robert J. Martin
Affiliation:
Chair for Nonlinear Analysis and Modeling, Faculty of Mathematics, University of Duisburg-Essen, Thea-Leymann-Str. 9, 45127, Essen, Germany Institute for Technologies of Metal, University of Duisburg-Essen, Friedrich-Ebert-Str. 12, 47119, Duisburg, Germany
Patrizio Neff
Affiliation:
Chair for Nonlinear Analysis and Modeling, Faculty of Mathematics, University of Duisburg-Essen, Thea-Leymann-Str. 9, 45127, Essen, Germany
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Abstract

According to a 2002 theorem by Cardaliaguet and Tahraoui, an isotropic, compact and connected subset of the group $\textrm {GL}^{\!+}(2)$ of invertible $2\times 2$ - - matrices is rank-one convex if and only if it is polyconvex. In a 2005 Journal of Convex Analysis article by Alexander Mielke, it has been conjectured that the equivalence of rank-one convexity and polyconvexity holds for isotropic functions on $\textrm {GL}^{\!+}(2)$ as well, provided their sublevel sets satisfy the corresponding requirements. We negatively answer this conjecture by giving an explicit example of a function $W\colon \textrm {GL}^{\!+}(2)\to \mathbb {R}$ which is not polyconvex, but rank-one convex as well as isotropic with compact and connected sublevel sets.

Keywords

quasiconvexityrank-one convexitypolyconvexitynonlinear elasticityhyperelasticityweak lower semi-continuityisotropic setscalculus of variations

MSC classification

Primary: 26B25: Convexity, generalizations
Secondary: 26A51: Convexity, generalizations 74B20: Nonlinear elasticity
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 152 , Issue 2 , April 2022 , pp. 356 - 381
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Copyright
Copyright © The Author(s), 2022. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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