In this paper we provide an estimate from above for the value of the relaxed area functional A¯(u, Ω) for an ℝ2-valued map u defined on a bounded domain Ω of the plane and discontinuous on a C2 simple curve J¯u ⊂ Ω, with two endpoints. We show that, under certain assumptions on u, A¯(u, Ω) 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 J¯u. 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.

Bellettini, G., Paolini, M., Tealdi, L. (2015). On the area of the graph of a piecewise smooth map from the plane to the plane with a curve discontinuity. ESAIM. COCV, 22(1), 29-63 [10.1051/cocv/2014065].

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

BELLETTINI, GIOVANNI;
2015-01-01

Abstract

In this paper we provide an estimate from above for the value of the relaxed area functional A¯(u, Ω) for an ℝ2-valued map u defined on a bounded domain Ω of the plane and discontinuous on a C2 simple curve J¯u ⊂ Ω, with two endpoints. We show that, under certain assumptions on u, A¯(u, Ω) 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 J¯u. 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.
2015
Bellettini, G., Paolini, M., Tealdi, L. (2015). On the area of the graph of a piecewise smooth map from the plane to the plane with a curve discontinuity. ESAIM. COCV, 22(1), 29-63 [10.1051/cocv/2014065].
File in questo prodotto:
File Dimensione Formato  
2015_Bellettini_Paolini_Tealdi_COCV.pdf

non disponibili

Tipologia: PDF editoriale
Licenza: NON PUBBLICO - Accesso privato/ristretto
Dimensione 845.5 kB
Formato Adobe PDF
845.5 kB Adobe PDF   Visualizza/Apri   Richiedi una copia

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11365/1017411