Abstract
In this paper we consider the anisotropic perimeter
defined on subsets E⊂ℝ2, where the anisotropy φ is a (possibly non-symmetric) norm on ℝ2 and ν E is the exterior unit normal vector to ∂E.
We consider quasi-minimal sets E (which include sets with prescribed curvature) and we prove that ∂E∖Σ(E) is locally a bi-Lipschitz curve and the singular set Σ(E) is closed and discrete.
We then classify the global Pφ-minimal sets. In particular we find that global minimal sets may have a singular point if and only if {φ≤1} is a triangle or a quadrilateral and that sets with two singularities exist if and only if {φ≤1} is a triangle.
We finally show that the boundary of a subset of ℝ2, which locally minimizes the anisotropic perimeter, plus a volume term (prescribed constant curvature) is contained, up to a translation and a rescaling, in the boundary of the Wulff shape determined by the anisotropy.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Almgren, F.J., Taylor, J., Wang, L.: Curvature-driven flows: a variational approach. SIAM J. Control Optimization 31, 387–437 (1993)
Ambrosio, L., Caselles, V., Masnou, S., Morel, J.-M.: Connected components of sets of finite perimeter and applications to image processing. J. Eur. Math. Soc. 3, 39–92 (2001)
Ambrosio, L., Fusco, N., Pallara, D.: Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs. Oxford: Clarendon Press 2000
Ambrosio, L., Novaga, M., Paolini, E.: Some regularity results for minimal crystals. ESAIM, Control Optim. Calc. Var. 8, 69–103 (2002)
Bellettini, G., Novaga, M., Paolini, M.: Facet-breaking for three-dimensional crystals evolving by mean curvature. Interfaces Free Bound. 1, 39–55 (1999)
Bombieri, E.: Regularity theory for almost minimal currents. Arch. Ration. Mech. Anal. 78, 99–130 (1982)
De Giorgi, E.: Nuovi teoremi relativi alle misure (r-1)-dimensionali in uno spazio a r dimensioni. Ric. Mat. 4, 95–113 (1955)
Finn, R.: Equilibrium capillary surfaces. New York: Springer 1986
Fonseca, I.: The wulff theorem revisited. Proc. R. Soc. Lond., Ser. A 432, 125–145 (1991)
Fonseca, I., Müller, S.: A uniqueness proof for the Wulff theorem. Proc. R. Soc. Edinb., Sect. A, Math. 119, 125–136 (1991)
Giusti, E.: Minimal surfaces and functions of bounded variation. In: Monographs in Mathematics Vol. 80. Birkhäuser 1984
Massari, U.: Esistenza e regolarità delle ipersuperfici di curvatura media assegnata in ℝn. Arch. Ration. Mech. Anal. 55, 357–382 (1974)
Morgan, F., French, C., Greenleaf, S.: Wulff clusters in ℝ2. J. Geom. Anal. 8, 97–115 (1998)
Pommerenke, C.: Boundary Behaviour of Conformal Maps. In: Grundlehren der Mathematischen Wissenschaften, Vol. 299. Springer 1992
Taylor, J.: Geometric crystal growth in 3D via facetted interfaces. In: Computational Crystal Growers Workshop, Selected Lectures in Mathematics, pp. 111–113. Am. Math. Soc. 1992
Yunger, J.: Facet Stepping and Motion By Crystalline Curvature. PhD thesis, Rutgers University, NJ 1998
Author information
Authors and Affiliations
Corresponding authors
Additional information
Mathematics Subject Classification (2000)
74N05, 49N60
Rights and permissions
About this article
Cite this article
Novaga, M., Paolini, E. Regularity results for boundaries in ℝ2 with prescribed anisotropic curvature. Annali di Matematica 184, 239–261 (2005). https://doi.org/10.1007/s10231-004-0112-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10231-004-0112-x