On the relaxed area of maps from the plane to itself taking three values

In this thesis we address two different problems; the first and the main one is the study of the relaxed area functional A(u, Ω) of the graph of vector maps u ∈ L1(Ω; R^k) on an open set Ω ⊂ R^n, mainly for n = k = 2. In particular we are interested in estimating 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 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. Moreover we investigate the possibility of adopting similar techniques to study the same problem in more general and different settings for instance when u has several (finite or infinite) triple junctions, or when n = 3, or when R^{2×2} is endowed by a Riemannian metric. An interesting open problem that we plan to address in the future, is to investigate the lower bound inequality; this seems to involve a strong use of geometric measure theory and Cartesian currents. The second part of the thesis is devoted to the problem of characterizing arbitrary codimensional smooth manifolds M with boundary embedded in R^n using the square distance function and the signed distance function from M and from its boundary. The results are localized in an open set.