Minimizers of anisotropic perimeters with cylindrical norms

TL;DR

This paper investigates the regularity and structure of minimizers of anisotropic perimeter functionals with cylindrical norms, establishing Lipschitz regularity, small singular sets, and a Bernstein-type characterization of entire minimizers.

Contribution

It provides new regularity results for anisotropic perimeter minimizers and proves a Bernstein-type theorem for entire minimizers depending on a single variable.

Findings

Boundary of minimizers is locally Lipschitz outside a small singular set.

Entire minimizers are subgraphs of monotone functions of one variable.

Singular sets have small Hausdorff dimension.

Abstract

We study various regularity properties of minimizers of the $\Phi$--perimeter, where $\Phi$ is a norm. Under suitable assumptions on $\Phi$ 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.