arXiv subjects
Publications and source records attributed to Luan de Figueiredo.
We prove a compactness theorem for the space of closed embedded minimal surfaces with area bounded from above and injectivity radius bounded from below in a closed Riemannian $3$-manifold. This result is a variant of the Choi--Schoen compactness theorem in which the genus bound is replaced by a lower bound on the injectivity radius of the surface.
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 $C^{1,α}$-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.
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 ($C^{1,α}$ in space and $C^{0,α/2}$ in time up to the free boundary) and $C^{1,α}$-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 $n - 1$.