Equivalence between rank-one convexity and polyconvexity for isotropic sets of \({\mathbb R}^{2{\times}2}\). II
As in the Part I of the paper [Nonlinear Analysis, Theory Methods Appl. 50A, No. 8, 1179-1199 (2002; Zbl 1004.49007, preceding review)], the authors consider closed subsets \(K\) of \(2\times 2\) matrices, classes of functions \[ {\mathcal E}(K)= \{u\in W^{1,\infty}(\Omega; \mathbb{R}^2):\nabla u(x)\in K\text{ for a.e. }x\in\Omega\} \] and deal with the problem of characterizing all sets \(K\) for which \({\mathcal E}(K)\) is weakly\(^*\) closed. This leads to study the notions of polyconvexity, quasiconvexity, rank-one convexity for subsets \(K\) of \(2\times 2\) matrices. The sets \(K\) considered in the Part I of the paper were of the form \[ K(\Lambda)= \{A\in \mathbb{R}^{2\times 2}: (\lambda_1(A), \lambda_2(A))\in \Lambda\} \] being \(\lambda_1(A)\) and \(\lambda_2(A)\) the singular values of the matrix \(A\) (i.e., the eigenvalues of \((AA^*)^{1/2}\)) and where \(\Lambda\subset E= \{(x,y)\in \mathbb{R}^2: 0\leq x\leq y\}\). In the present Part II they add the constraint \(\text{det }A\geq 0\), which gives the sets \[ K_+(\Lambda)= \{A\in K(\Lambda): \text{det }A\geq 0\}. \] The main result (Theorem 7.1) is that when \(\Lambda\subset E\) is compact and connected, then \(K_+(\Lambda)\) is polyconvex if and only if it is rank-one convex. Though formally similar to the results of Part I (except for the connectedness assumption that was not required in the unconstrained case), this constrained case seems more difficult and several new tools are developed in the paper.
scientific article; zbMATH DE number 1788065
scientific article