A Halfspace Theorem for Global Anisotropic Perimeter Minimizers

Abstract

We prove an arbitrary-dimensional halfspace theorem for global minimizers of a smooth uniformly elliptic anisotropic perimeter: if the nonempty boundary of a minimizing set is contained in a halfspace, then the set itself is a halfspace. No evenness of the integrand or regularity of the minimizing boundary is assumed. The proof combines wall contact with a plane-peeling argument for the complementary phase defect and a simultaneous second blow-down. We also give a direct, self-contained exterior-barrier proof of the two-dimensional stationary anisotropic statement. Its rigidity conclusion is already covered by Bergner's earlier halfspace theorem; the proof is retained because it works directly with the anisotropic Euler-Lagrange operator.

23 pages, 5 figures. v2: Added Bergner's 2010 reference and clarified that the two-dimensional rigidity statement was previously known; the direct exterior-barrier proof is retained