Regularity results for boundaries in R2 with prescribed anisotropic curvature

In this paper we consider the anisotropic perimeter Pϕ(E)=∫∂Eφ(νE)dH1 defined on subsets E⊂R2, where the anisotropy φ is a (possibly non symmetric) norm on R2 and νE 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.