Mathematics

In this talk 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 $C^2-$ embedded curves that meet transversely at only one point. We show  that the relaxed area can be estimated from above by the solution of a Plateau-type problem  involving three entangled nonparametric area-minimizing surfaces. The construction of the surfaces depends on the choice of a target triple junction, and 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 that what infimizing over all possible target triple junctions and all corresponding connections.