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arXiv · 1103.2859

Young measures supported on invertible matrices

Barbora Benešová·Martin Kružík·Gabriel Pathó
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Abstract

Motivated by variational problems in nonlinear elasticity depending on the deformation gradient and its inverse, we completely and explicitly describe Young measures generated by matrix-valued mappings $\{Y_k\}_{k\in\N} \subset L^p(Ø;\R^{n\times n})$, $Ø\subset\R^n$, such that $\{Y_k^{-1}\}_{k\in\N} \subset L^p(Ø;\R^{n\times n})$ is bounded, too. Moreover, the constraint $\det Y_k>0$ can be easily included and is reflected in a condition on the support of the measure. This condition typically occurs in problems of nonlinear-elasticity theory for hyperelastic materials if $Y:=\nabla y$ for $y\in W^{1,p}(Ø;\R^n)$. Then we fully characterize the set of Young measures generated by gradients of a uniformly bounded sequence in $W^{1,\infty}(Ø;\R^n)$ where the inverted gradients are also bounded in $L^\infty(Ø;\R^{n\times n})$. This extends the original results due to D. Kinderlehrer and P. Pedregal.

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Barbora Benešová, Martin Kružík, Gabriel Pathó. 2013-01-17. Young measures supported on invertible matrices. https://arxiv.org/abs/1103.2859

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