Calculus of Variations and Geometric Measure Theory

M. Novaga - E. Paolini

Regularity results for boundaries in ${\mathrm R}^2$ with prescribed anisotropic curvature

created on 07 Oct 2000
modified by root on 18 Mar 2012


Published Paper

Inserted: 7 oct 2000
Last Updated: 18 mar 2012

Journal: Annali di Matematica pura e Applicata
Volume: 184
Number: 2
Pages: 239-261
Year: 2005


In this paper we consider the anisotropic perimeter \[ P_\phi(E)=\int_{\partial E} \varphi(\nu_E)d \mathcal H^1 \] defined on subsets $E\subset \mathbb R^2$, where the anisotropy $\varphi$ is a (possibly non symmetric) norm on $\mathbb R^2$ and $\nu_E$ is the exterior unit normal vector to $\partial E$.

We consider quasi-minimal sets $E$ (which include sets with prescribed curvature) and we prove that $\partial E\setminus\Sigma(E)$ is locally a bi-lipschitz curve and the singular set $\Sigma(E)$ is closed and discrete.

We then classify the global $P_\varphi$-minimal sets. In particular we find that global minimal sets may have a singular point if and only if $\{\varphi\le 1\}$ is a triangle or a quadrilateral and that sets with two singularities exist if and only if $\{\varphi\le 1\}$ is a triangle.

We finally show that the boundary of a subset of $\mathbb R^2$ 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.

Keywords: regularity, perimeter, crystals