Anisotropic Obstacle Problems for Minimal Surfaces: Regularity of the Free Boundary via the Cahn-Hoffman Transform

Abstract

We study an obstacle problem for surfaces minimizing an anisotropic surface energy of ellipsoidal type. Given a convex obstacle and a boundary datum, we seek a surface that minimizes the anisotropic area functional while remaining above the obstacle. The central novelty is the systematic use of the Cahn-Hoffman transform to convert the anisotropic problem into an equivalent isotropic problem with a generalized Robin boundary condition. We prove optimal regularity of the solution ($C^{1,1}$ up to the free boundary) and $C^{1,\alpha }$-regularity of the free boundary itself under a non-degeneracy condition. The singular set of the free boundary is shown to have Hausdorff dimension at most $n-1$, and a logarithmic epiperimetric inequality yields its $(n-1)$-rectifiability. The approach combines Caffarelli's classical theory of obstacle problems with the geometric theory of anisotropic mean curvature and the Alexandrov reflection principle adapted to the anisotropic setting.