Characterization of gradient Young measures generated by homeomorphisms in the plane

We characterize Young measures generated by gradients of bi-Lipschitz orientation-preserving maps in the plane. This question is motivated by variational problems in nonlinear elasticity where the orientation preservation and injectivity of the admissible deformations are key requirements. These results enable us to derive new weak$^*$ lower semicontinuity results for integral functionals depending on gradients. As an application, we show the existence of a minimizer for an integral functional with nonpolyconvex energy density among bi-Lipschitz homeomorphisms.Comment: ESAIM Control Optim. Calc. Va

Young measures supported on invertible matrices

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

A note on equality of functional envelopes

summary:We characterize generalized extreme points of compact convex sets. In particular, we show that if the polyconvex hull is convex in $\mathbb{R}^{m\times n}$, $\min (m,n)\le 2$, then it is constructed from polyconvex extreme points via sequential lamination. Further, we give theorems ensuring equality of the quasiconvex (polyconvex) and the rank-1 convex envelopes of a lower semicontinuous function without explicit convexity assumptions on the quasiconvex (polyconvex) envelope