Calculus of Variations and Geometric Measure Theory

G. Bellettini - M. Paolini - L. Tealdi

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

created by tealdi on 08 Oct 2013
modified on 24 Mar 2015


Accepted Paper

Inserted: 8 oct 2013
Last Updated: 24 mar 2015

Year: 2014
Doi: ESAIM: Control, Optimisation and Calculus of Variation


In this paper we provide an estimate from above for the value of the relaxed area functional $\overline{\mathcal{A}}({\bf u}, \Omega)$ for an $\mathbb{R}^2$-valued map ${\bf u}$ defined on a bounded domain $\Omega$ of the plane and discontinuous on a $\mathcal{C}^2$ simple curve $J_{\bf u}\subset \Omega$, with two endpoints. We show that, under certain assumptions on ${\bf u}$, $\overline{\mathcal{A}}({\bf u}, \Omega)$ does not exceed the area of the regular part of ${\bf u}$, with the addition of a singular term measuring the area of a disk-type solution $\Sigma_{\mbox{min}}$ of the Plateau’s problem spanning the two traces of ${\bf u}$ on $J_{\bf u}$. The result is valid also when $\Sigma_{\mbox{min}}$ has self-intersections. A key element in our argument is to show the existence of what we call a semicartesian parametrization of $\Sigma_{\mbox{min}}$ , namely a conformal parametrization of $\Sigma_{\mbox{min}}$ 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