T5-Configurations and non-rigid sets of matrices
T5-Configurations and non-rigid sets of matrices
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DOI：
10.1007/s00526-017-1293-7
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发表时间：
2018-02-01
影响因子：
2.1
通讯作者：
Szekelyhidi, Laszlo, Jr.
中科院分区：
文献类型：
--
作者：
Foerster, Clemens;Szekelyhidi, Laszlo, Jr.
2003年B。基希海姆-D Preiss在平面上构造了一个具有5个不相容梯度的Lipschitz映射，其中不相容是指5个矩阵中没有两个是秩1连通的。该结构是通过凸积分的方法，并依赖于从一组特定的五个矩阵的秩一几何的详细理解。2010年，W. Pompe in Calc.变种PDE 37（3-4）：461-473，（2010）通过精细的几何论证。对于更一般的矩阵集合，秩一凸船体的完整计算显然是遥不可及的。因此，在这个简短的说明，我们重新建设，并提出了一个新的，在某种意义上通用的方法来决定是否可以进行一组给定的矩阵的凸积分，这不需要完整的计算秩一凸船体。
In 2003 B. Kirchheim-D. Preiss constructed a Lipschitz map in the plane with 5 incompatible gradients, where incompatibility refers to the condition that no two of the five matrices are rank-one connected. The construction is via the method of convex integration and relies on a detailed understanding of the rank-one geometry resulting from a specific set of five matrices. The full computation of the rank-one convex hull for this specific set was later carried out in 2010 by W. Pompe in Calc. Var. PDE 37(3-4): 461-473, (2010) by delicate geometric arguments. For more general sets of matrices a full computation of the rank-one convex hull is clearly out of reach. Therefore, in this short note we revisit the construction and propose a new, in some sense generic method for deciding whether convex integration for a given set of matrices can be carried out, which does not require the full computation of the rank-one convex hull.