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Automatic Quasiconvexity of Homogeneous Isotropic Rank-One Convex Integrands

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Abstract

We consider the class of non-negative rank-one convex isotropic integrands on \(\mathbb {R}^{n\times n}\) which are also positively p-homogeneous. If \(p\le n=2\) we prove, conditional on the quasiconvexity of the Burkholder integrand, that the integrands in this class are quasiconvex at conformal matrices. If \(p\ge n =2\), we show that the positive part of the Burkholder integrand is polyconvex. In general, for \(p\ge n\), we prove that the integrands in the above class are polyconvex at conformal matrices. Several examples imply that our results are all nearly optimal.

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Acknowledgements

A.G. was supported by the Infosys Membership at the Institute for Advanced Study. It is a pleasure to thank Kari Astala and Daniel Faraco for interesting discussions.

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Authors and Affiliations

  1. School of Mathematics, Institute for Advanced, Study, 1 Einstein Dr., Princeton, NJ, 08540, USA

    André Guerra

  2. University of Oxford, Andrew Wiles Building Woodstock Rd, Oxford, OX2 6GG, UK

    Jan Kristensen

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  1. André Guerra
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  2. Jan Kristensen
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Guerra, A., Kristensen, J. Automatic Quasiconvexity of Homogeneous Isotropic Rank-One Convex Integrands. Arch Rational Mech Anal 245, 479–500 (2022). https://doi.org/10.1007/s00205-022-01792-2

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