*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

**Abstract:**

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

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