Equivalence between rank-one convexity and polyconvexity for isotropic sets of \({\mathbb R}^{2{\times}2}\). I
Given a closed subset \(K\) of \(\mathbb{R}^{n\times n}\), the set of all \(n\times n\) matrices, the authors consider the class of functions \[ {\mathcal E}(K)= \{u\in W^{1,\infty}(\Omega; \mathbb{R}^n:\nabla u(x)\in K\text{ for a.e. }x\in \Omega\}. \] Here \(\Omega\) is an open subset of \(\mathbb{R}^n\). The question is to characterize the subsets \(K\) for which \({\mathcal E}(K)\) is weakly\(^*\) closed. It is known that: (i) if \({\mathcal E}(K)\) is weakly\(^*\) closed, then \(K\) is rank-one convex; (ii) if \(K\) is quasiconvex, then \({\mathcal E}(K)\) is weakly\(^*\) closed. The related questions of characterizing the weak\(^*\) closure of \({\mathcal E}(K)\) and correspondingly some polyconvex/quasiconvex/rank-one convex envelope of \(K\) arises. In the paper the case \(n=2\) of \(2\times 2\) matrices is considered. In particular, if \(E= \{(x,y)\in \mathbb{R}^2: 0\leq x\leq y\}\), the authors consider only sets \(K\) of the form \[ K(\Lambda)= \{A\in \mathbb{R}^{2\times 2}: (\lambda_1(A), \lambda_2(A))\in \Lambda\}, \] where \(\Lambda\subset E\) and \(\lambda_1(A)\), \(\lambda_2(A)\) denote the singular values of the matrix \(A\) (i.e., the eigenvalues of \((AA^*)^{1/2})\). The main result (Theorem 5.1) is that when \(\Lambda\subset E\) is compact, then \(K(\Lambda)\) is polyconvex if and only if it is rank-one convex.
scientific article; zbMATH DE number 1788064
scientific article