TY - JOUR

T1 - On the area of the graph of a piecewise smooth map from the plane to the plane with a curve discontinuity

AU - Bellettini, Giovanni

AU - Paolini, Maurizio

AU - Tealdi, Lucia

PY - 2016

Y1 - 2016

N2 - In this paper we provide an estimate from above for the value of the relaxed area functional for a map defined on a bounded domain Ω of the plane taking values in the real plane and discontinuous on a simple curve, with two endpoints. We show that, under certain assumptions, the relaxed area does not exceed the area of the regular part of the function, with the addition of a singular term measuring the area of a disk-type solution of the Plateau's problem spanning the two traces of the function across the jump set. The result is valid also when the minimal surface has self-intersections. A key element in our argument is to show the existence of what we call a semicartesian parametrization of the minimal surface, namely a conformal parametrization 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.

AB - In this paper we provide an estimate from above for the value of the relaxed area functional for a map defined on a bounded domain Ω of the plane taking values in the real plane and discontinuous on a simple curve, with two endpoints. We show that, under certain assumptions, the relaxed area does not exceed the area of the regular part of the function, with the addition of a singular term measuring the area of a disk-type solution of the Plateau's problem spanning the two traces of the function across the jump set. The result is valid also when the minimal surface has self-intersections. A key element in our argument is to show the existence of what we call a semicartesian parametrization of the minimal surface, namely a conformal parametrization 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.

KW - area of a graph

KW - codimension two

KW - area of a graph

KW - codimension two

UR - http://hdl.handle.net/10807/72398

U2 - 10.1051/cocv/2014065

DO - 10.1051/cocv/2014065

M3 - Article

SN - 1292-8119

VL - 22

SP - 29

EP - 63

JO - ESAIM. COCV

JF - ESAIM. COCV

ER -