Young measures supported on invertible matrices

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 $\{Y_k{\}}_{k\in \mathbb{N}}\subset L^p(Ø;{\mathbb{R}}^{n\times n})$, $Ø\subset {\mathbb{R}}^n$, such that $\{Y_k^{-1}{\}}_{k\in \mathbb{N}}\subset L^p(Ø;{\mathbb{R}}^{n\times n})$ 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 $y\in W^{1,p}(Ø;{\mathbb{R}}^n)$. Then we fully characterize the set of Young measures generated by gradients of a uniformly bounded sequence in $W^{1,\infty }(Ø;{\mathbb{R}}^n)$ where the inverted gradients are also bounded in $L^{\infty }(Ø;{\mathbb{R}}^{n\times n})$. This extends the original results due to D. Kinderlehrer and P. Pedregal.