论文标题
$ \ operatatorName {gl}^+（2）$带有紧凑的连接sublevel Sets上的$ \ operatatorName上的等级一凸，非polyconvex
A rank-one convex, non-polyconvex isotropic function on $\operatorname{GL}^+(2)$ with compact connected sublevel sets
论文作者
论文摘要
根据Cardaliaguet和Tahraoui的2002定理，该组的各向同性，紧凑且连接的子集$ \ permatatorName {gl}^+（2）可转让$ 2 \ times2 - \，$矩阵的$仅是rank-One convex if and pollyConconvex。在亚历山大·米尔克（Alexander〜Mielke）的2005年《凸分析杂志》（Journal of Alexander〜Mielke）的一篇文章中，人们猜想，等级 - 一凸和多峰的等效性在$ \ operatotorname {gl}^+（2）$上也适用于各向同性函数，只要他们的超级平台满足了相应的要求。我们通过给出一个函数$ w的明确示例$ w：\ operatorname {gl}^+\ to \ mathbb {r} $，而不是polyconvex，而是rank-One凸面以及与紧凑型和连接的Sublevel集合。
According to a 2002 theorem by Cardaliaguet and Tahraoui, an isotropic, compact and connected subset of the group $\operatorname{GL}^+(2)$ of invertible $2\times2-\,$matrices is rank-one convex if and only if it is polyconvex. In a 2005 Journal of Convex Analysis article by Alexander~Mielke, it has been conjectured that the equivalence of rank-one convexity and polyconvexity holds for isotropic functions on $\operatorname{GL}^+(2)$ as well, provided their sublevel sets satisfy the corresponding requirements. We negatively answer this conjecture by giving an explicit example of a function $W:\operatorname{GL}^+\to\mathbb{R}$ which is not polyconvex, but rank-one convex as well as isotropic with compact and connected sublevel sets.
