Anisotropic Parabolic Obstacle Problems and the Stefan Problem: Regularity of the Evolving Free Boundary

Abstract

We study a parabolic obstacle problem for surfaces evolving by anisotropic mean curvature flow subject to an obstacle constraint. Given a convex obstacle and initial data, we seek an evolving surface minimizing an anisotropic energy functional while remaining above the obstacle; as a special case, this framework includes the anisotropic Stefan problem, where the free boundary represents a phase transition interface with direction-dependent surface tension. The central tool is the Cahn--Hoffman transform $S(x) = A^{-1/2}x$, which maps the Wulff ellipsoid $\{x : x^T A^{-1}x \leq 1\}$ to the Euclidean unit ball and converts the anisotropic problem into an equivalent isotropic one with a generalized Robin-type condition on the free boundary. We prove optimal regularity of the solution ($C^{1,\alpha}$ in space and $C^{0,\alpha/2}$ in time up to the free boundary) and $C^{1,\alpha}$-regularity of the evolving free boundary at non-degenerate points. The parabolic Hausdorff dimension of the space-time singular set is shown to be at most $n - 1$.