关于平面上极小簇的 Steiner 性质。各向异性情形
On the Steiner property for planar minimizing clusters. The anisotropic case
AI总结：
本文研究了具有各向异性双重密度的平面上极小簇的 Steiner 性质，证明了在密度非常弱的假设下，极小簇仍由有限多条 C^{1,γ} 弧在有限多个三重点相交组成，且三重点处的夹角不一定为 120°。
摘要：
在本文中，我们讨论了具有各向异性双重密度的平面上极小簇的 Steiner 性质。这意味着我们考虑了经典的簇等周问题，但体积和周长是通过使用两种密度来定义的。特别地，周长密度也可能依赖于法向量的方向。对于 Euclid 情形（对应于两种密度均等于 $1$）的经典“Steiner 性质”表明，极小簇由有限多条 ${\rm C}^{1,γ}$ 弧组成，并在有限多个“三重点”处相交。我们可以证明，在对密度的非常弱的假设下，该性质仍然成立。在平行的论文《关于平面上极小簇的 Steiner 性质。各向同性情形》中，我们考虑了各向同性情形，即周长密度不依赖于方向，这使得大部分构造更为简单。特别地，在当前情形下，三重点处的三条弧不一定以三个 $120^\circ$ 的角相交，而这正是各向同性情形中所发生的情况。
英文摘要：
In this paper we discuss the Steiner property for minimal clusters in the plane with an anisotropic double density. This means that we consider the classical isoperimetric problem for clusters, but volume and perimeter are defined by using two densities. In particular, the perimeter density may also depend on the direction of the normal vector. The classical "Steiner property" for the Euclidean case (which corresponds to both densities being equal to $1$) says that minimal clusters are made by finitely many ${\rm C}^{1,γ}$ arcs, meeting in finitely many "triple points". We can show that this property holds under very weak assumptions on the densities. In the parallel paper "On the Steiner property for planar minimizing clusters. The isotropic case" we consider the isotropic case, i.e., when the perimeter density does not depend on the direction, which makes most of the construction much simpler. In particular, in the present case the three arcs at triple points do not necessarily meet with three angles of $120^\circ$, which is instead what happens in the isotropic case.
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