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

Bellettini, Giovanni; Paolini, Maurizio; Tealdi, Lucia

ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 1, pp. 29-63.

In this paper we provide an estimate from above for the value of the relaxed area functional $(,\Omega )$ for an ${ℝ}^2$-valued map $𝐮$ defined on a bounded domain $\Omega$ of the plane and discontinuous on a ${ℂ}^2$ simple curve $\subset$, with two endpoints. We show that, under certain assumptions on $𝐮$, $(,\Omega )$ 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 $\Sigma {}_{min}$ of the Plateau’s problem spanning the two traces of $𝐮$ on $$. The result is valid also when $\Sigma {}_{min}$ has self-intersections. A key element in our argument is to show the existence of what we call a semicartesian parametrization of $\Sigma {}_{min}$, namely a conformal parametrization of $\Sigma {}_{min}$ 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.

E. Acerbi and G. Dal Maso, New lower semicontinuity results for polyconvex integrals. Calc. Var. Partial Differ. Eqs. 3 (1994) 329–371. | DOI | Zbl

G. Bellettini and M. Paolini, On the area of the graph of a singular map from the plane to the plane taking three values. Adv. Calc. Var. 3 (2010) 371–386. | DOI | Zbl

B. Dacorogna, Direct Methods in the Calculus of Variations. Springer-Verlag, Berlin (1989). | Zbl

E. De Giorgi, On the relaxation of functionals defined on cartesian manifolds, in Developments in Partial Differential Equations and Applications in Mathematical Physics (Ferrara 1992). Plenum Press, New York (1992) 33–38. | Zbl

U. Dierkes, S. Hildebrandt and F. Sauvigny, Minimal Surfaces. Vol. 339 of Grundlehren der Mathematischen. Springer, Berlin (2010). | Zbl

U. Dierkes, S. Hildebrandt and A. Tromba, Regularity of Minimal Surfaces. Vol. 340 of Grundlehren der Mathematischen. Springer, Berlin (2010). | Zbl

M. Giaquinta, G. Modica and J. Soucek, Cartesian Currents in the Calculus of Variations. Springer-Verlag, Berlin (1998). | Zbl

E. Giusti, Minimal Surfaces and Functions of Bounded Variation. Birkhäuser, Boston (1984). | Zbl

R. Gulliver, F.D. Lesley, On boundary branch points of minimizing surface. Arch. Ration. Mech. Anal. 52 (1973) 20–25. | DOI | Zbl

H. Lewy, On the boundary of minimal surfaces. Proc. Natl. Acad. Sci USA 37 (1951) 103–110. | DOI | Zbl

F. Morgan, Geometric Measure Theory. A Beginner’s Guide. Academic Press, Inc. Boston (1988). | Zbl

U. Massari and M. Miranda, Minimal Surfaces of Codimension One. Amsterdam, North-Holland (1984). | Zbl

M. Morse and G.B. Van Schaack, The critical point theory under general boundary condition. Ann. Math. 35 (1934) 545–571. | DOI | JFM

J.C.C. Nitsche, Lectures on Minimal Surfaces. Cambridge University Press, Cambridge (1989)

R. Osserman, A proof of the regularity everywhere of the classical solution to Plateau’s problem. Ann. Math. 91 (1970) 550–569. | DOI | Zbl