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,
Italy, and INFN Laboratori
Nazionali di Frascati, Frascati, Italy.
2 Dipartimento di Matematica, Università Cattolica “Sacro Cuore”, via Trieste 17, 25121 Brescia, Italy.
3 International School for Advanced Studies, S.I.S.S.A., via Bonomea 265, 34136 Trieste, Italy.
Revised: 14 November 2014
In this paper we provide an estimate from above for the value of the relaxed area functional for an ℝ2-valued map u defined on a bounded domain Ω of the plane and discontinuous on a simple curve , with two endpoints. We show that, under certain assumptions on u, 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 Σmin of the Plateau’s problem spanning the two traces of u on . The result is valid also when Σmin has self-intersections. A key element in our argument is to show the existence of what we call a semicartesian parametrization of Σmin, namely a conformal parametrization of Σmin 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.
Mathematics Subject Classification: 49J45 / 49Q05
Key words: Relaxation / area functional in codimension two / disk-type minimal surfaces / Plateau’s problem
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