条纹图案和程函方程
Stripe patterns and the eikonal equation
摘要：
研究程函方程的一种新形式|gradu|在R^2的有界子集上，考虑一个在1维子空间上的正交投影场P，且散度在L^2内有界，我们证明了方程P div P=0的解的存在唯一性。我们给出了一个几何描述，与经典的情况下，我们证明，这样的解决方案存在，只有当域是一个经常闭曲线的管状邻域。该公式为分析条纹图案提供了一种有用的方法。它特别适用于图案的物理性质在180度旋转下不变的系统，例如嵌段共聚物或液晶系统。
英文摘要：
We study a new formulation for the eikonal equation |grad u| =1 on a bounded subset of R^2. Considering a field P of orthogonal projections onto 1-dimensional subspaces, with divergence bounded in L^2, we prove existence and uniqueness for solutions of the equation P div P=0. We give a geometric description, comparable with the classical case, and we prove that such solutions exist only if the domain is a tubular neighbourhood of a regular closed curve. This formulation provides a useful approach to the analysis of stripe patterns. It is specifically suited to systems where the physical properties of the pattern are invariant under rotation over 180 degrees, such as systems of block copolymers or liquid crystals.
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