On the area of the graph of a piecewise smooth map from the plane to the plane with a curve discontinuity

¹ Dipartimento di Matematica, Università di Roma Tor Vergata, via della Ricerca Scientifica 1, 00133 Roma, Italy, and INFN Laboratori Nazionali di Frascati, Frascati, Italy.
belletti@mat.uniroma2.it
² Dipartimento di Matematica, Università Cattolica “Sacro Cuore”, via Trieste 17, 25121 Brescia, Italy.
paolini@dmf.unicatt.it
³ International School for Advanced Studies, S.I.S.S.A., via Bonomea 265, 34136 Trieste, Italy.
ltealdi@sissa.it

Received: 3 March 2014
Revised: 14 November 2014

Abstract

In this paper we provide an estimate from above for the value of the relaxed area functional $\overline{\mathrm{𝒜}}\mathrm{\left(}{u}\mathit{,\Omega }\mathrm{\right)}$ for an ℝ²-valued map u defined on a bounded domain Ω of the plane and discontinuous on a simple curve ${\overline{\mathit{J}}}_{{u}}\mathrm{\subset }\mathit{\Omega }$, with two endpoints. We show that, under certain assumptions on u, $\overline{\mathrm{𝒜}}\mathrm{\left(}{u}\mathit{,\Omega }\mathrm{\right)}$ does not exceed the area of the regular part of u, with the addition of a singular term measuring the area of a disk-type solution Σ_min of the Plateau’s problem spanning the two traces of u on ${\overline{\mathit{J}}}_{{u}}$. The result is valid also when Σ_min has self-intersections. A key element in our argument is to show the existence of what we call a semicartesian parametrization of Σ_min, namely a conformal parametrization of Σ_min defined on a suitable parameter space, which is the identity in the first component. To prove our result, various tools of parametric minimal surface theory are used, as well as some results from Morse theory.