# On the relaxed area of the graph of discontinuous maps from the plane to the plane taking three values with no symmetry assumptions

## TL;DR

This paper estimates the relaxed area of a discontinuous map from a disk to three points in the plane, using a Plateau-type problem involving three minimal surfaces coupled at a triple point.

## Contribution

It introduces a novel method to bound the relaxed area by relating it to a minimal surface problem with a triple junction, without symmetry assumptions.

## Key findings

- Relaxed area is bounded above by a Plateau-type problem solution.
- Construction of smooth approximations via three coupled minimal surfaces.
- Dependence of the estimate on the choice of target triple point and connection.

## Abstract

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 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.