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

Maurizio Paolini, Giovanni Bellettini, Alaa Elshorbagy, Riccardo Scala

Research output: Contribution to journalArticlepeer-review

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 C2-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.
Original languageEnglish
Pages (from-to)1-33
Number of pages33
JournalAnnali di Matematica Pura ed Applicata
DOIs
Publication statusPublished - 2019

Keywords

  • Area functional
  • Cartesian currents
  • Minimal surfaces
  • Plateau problem
  • Relaxation

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