In this paper, we estimate from above the area of the graph of a singular map u taking a disk to three vectors, the vertices of a triangle, and jumping along three \({\mathcal {C}}^2\)-embedded curves that meet transversely at only one point of the disk. We show that the singular part of the relaxed area can be estimated from above by the solution of a Plateau-type problem involving three entangled nonparametric area-minimizing surfaces. The idea is to “fill the hole” in the graph of the singular map with a sequence of approximating smooth two-codimensional surfaces of graph-type, by imagining three minimal surfaces, placed vertically over the jump of u, coupled together via a triple point in the target triangle. Such a construction depends on the choice of a target triple point, and on a connection passing through it, which dictate the boundary condition for the three minimal surfaces. We show that the singular part of the relaxed area of u cannot be larger than what we obtain by minimizing over all possible target triple points and all corresponding connections.

在本文中，我们从一个奇异地图的曲线图的区域上方估计ü采取盘三个矢量，一个三角形的顶点，和跳跃沿三个\（{\ mathcal {C}} ^ 2 \） -嵌入式在圆盘的仅一点处横向相交的曲线。我们表明，可以通过涉及三个纠缠的非参数面积最小化曲面的高原型问题的解，从上方估计松弛区域的奇异部分。这个想法是通过想象三个最小的，垂直于u跃迁的最小表面，用一系列近似的图型光滑二维表面“填充”奇异图的图形中的孔。，通过目标三角形中的三点耦合在一起。这种构造取决于目标三点的选择以及通过它的连接，这决定了三个最小曲面的边界条件。我们证明u的松弛区域的奇异部分不能大于我们通过最小化所有可能的目标三点和所有对应的连接所获得的部分。