Regularity results for boundaries in R^2 with prescribed anisotropic curvature

In this paper we consider the anisotropic perimeter P-phi (E) = integral(partial derivative E) phi(nu(E)) dH(1) defined on subsets E subset of R-2, where the anisotropy phi is a (possibly non-symmetric) norm on R-2 and nu(E) is the exterior unit normal vector to partial derivative E. We consider quasi-minimal sets E (which include sets with prescribed curvature) and we prove that partial derivative E \ Sigma(E) is locally a bi-Lipschitz curve and the singular set Sigma(E) is closed and discrete. We then classify the global P-phi-minimal sets. In particular we find that global minimal sets may have a singular point if and only if {phi <= 1} is a triangle or a quadrilateral and that sets with two singularities exist if and only if {phi <= 1} is a triangle. We finally show that the boundary of a subset of 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.