Critical points of degenerate polyconvex energies

We study critical and stationary, i.e. critical with respect to both inner and outer variations, points of polyconvex functionals of the form \(f(X) = g(\det(X))\), for \(X \in \mathbb{R}^{2\times 2}\). In particular, we show that critical points \(u \in Lip(\Omega,\mathbb{R}^2)\) with \(\det(Du) \n...

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Bibliographic Details
Published inarXiv.org
Main Author Tione, Riccardo
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 23.03.2022
Subjects
Critical point
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Summary:We study critical and stationary, i.e. critical with respect to both inner and outer variations, points of polyconvex functionals of the form \(f(X) = g(\det(X))\), for \(X \in \mathbb{R}^{2\times 2}\). In particular, we show that critical points \(u \in Lip(\Omega,\mathbb{R}^2)\) with \(\det(Du) \neq 0\) a.e. have locally constant determinant except in a relatively closed set of measure zero, and that stationary points have constant determinant almost everywhere. This is deduced from a more general result concerning solutions \(u \in Lip(\Omega,\mathbb{R}^n)\), \(\Omega \subset \mathbb{R}^n\) to the linearized problem \(curl(\beta Du) = 0\). We also present some generalization of the original result to higher dimensions and assuming further regularity on solutions \(u\). Finally, we show that the differential inclusion associated to stationarity with respect to polyconvex energies as above is rigid.
ISSN:2331-8422