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 {\textquoteleft}{\textquoteleft}fill the hole{\textquoteright}{\textquoteright} 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.

}, issn = {1618-1891}, doi = {10.1007/s10231-019-00887-0}, url = {https://doi.org/10.1007/s10231-019-00887-0}, author = {Giovanni Bellettini and Alaa Elshorbagy and Maurizio Paolini and Riccardo Scala} } @article {2019, title = {On the square distance function from a manifold with boundary}, year = {2019}, abstract = {We characterize arbitrary codimensional smooth manifolds $\mathcal{M}$ with boundary embedded in $\mathbb{R}^n$ using the square distance function and the signed distance function from $\mathcal{M}$ and from its boundary. The results are localized in an open set.

}, url = {http://cvgmt.sns.it/media/doc/paper/4260/manif_with_bound_dist.pdf}, author = {Giovanni Bellettini and Alaa Elshorbagy} }