Minimizers of anisotropic perimeters with cylindrical norms

We study various regularity properties of minimizers of the $\mathrm{\Phi }$–perimeter, where $\mathrm{\Phi }$ is a norm. Under suitable assumptions on $\mathrm{\Phi }$ and on the dimension of the ambient space, we prove that the boundary of a cartesian minimizer is locally a Lipschitz graph out of a closed singular set of small Hausdorff dimension. Moreover, we show the following anisotropic Bernstein-type result: any entire cartesian minimizer is the subgraph of a monotone function depending only on one variable.