@article {Bellettini2019,
	title = {On the relaxed area of the graph of discontinuous maps from the plane to the plane taking three values with no symmetry assumptions},
	journal = {Annali di Matematica Pura ed Applicata (1923 -)},
	year = {2019},
	month = {Jul},
	abstract = {<p>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.</p>},
	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 {2015,
	title = {Results on the minimization of the Dirichlet functional among semicartesian parametrizations},
	year = {2015},
	note = {The article is compsed of 18 pages and is recorded in PDF format},
	abstract = {<p>We start to investigate the existence of conformal minimizers for the Dirichlet functional in the setting of the so-called semicartesian parametrizations, adapting to this context some techniques used in solving the classical Plateau{\textquoteright}s problem. The final goal is to find area minimizing semicartesian parametrizations spanning a Jordan curve obtained as union of two graphs; this problem appeared in the study of the relaxed area functional for maps from the plane to the plane jumping on a line.</p>},
	url = {http://urania.sissa.it/xmlui/handle/1963/34488},
	author = {Lucia Tealdi and Giovanni Bellettini and Maurizio Paolini}
}
