The Failure of Rank-One Connections
The Failure of Rank-One Connections
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DOI：
10.1007/s002050200197
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发表时间：
2002
影响因子：
2.5
通讯作者：
A. Vogel
中科院分区：
文献类型：
--
作者：
T. Iwaniec;G. Verchota;A. Vogel
本文研究了半空间中Lipschitz映射f+：n+= n−上一致.这些自然发生在材料微观结构和晶体的数学模型中。任务是确定微分Df+和Df−的值集合之间的关系。一段时间以来，人们一直认为多凸壳[Df+]pc和[Df-]pc满足Hadamard跳跃条件或至少是秩一连通的。我们在这里的例子反驳了这个想法，我们在解决所谓的蒙赫-安培不等式的过程中得到的雅可比行列式的估计似乎也是独立的兴趣。作为应用，我们构造了与熟悉的Cauchy-Riemann方程在同一同伦类中的一阶偏微分方程的一致椭圆型方程组，其唯一的连续性不成立.
Abstract This article is concerned with interface problems for Lipschitz mappings f+:ℝn+→ℝn and f−:ℝn−→ℝn in the half spaces, which agree on the common boundary ℝn− 1=∂ℝn+=∂ℝn−. These naturally occur in mathematical models for material microstructures and crystals. The task is to determine the relationship between the sets of values of the differentials Df+ and Df−. For some time it has been thought that the polyconvex hulls [Df+]pc and [Df−]pc satisfy Hadamard's jump condition or are at least rank-one connected. Our examples here refute this idea.The estimates of the Jacobians we obtain in the course of solving the so-called Monge-Ampère inequalities seem also to be of independent interest. As an application, we construct uniformly elliptic systems of first order partial differential equations in the same homotopy class as the familiar Cauchy-Riemann equations, for which the unique continuation property fails.