Regularity of minimal surfaces with lower-dimensional obstacles

Abstract

AbstractWe study the Plateau problem with a lower-dimensional obstacle in {\mathbb{R}^{n}}. Intuitively, in {\mathbb{R}^{3}} this corresponds to a soap film (spanning a given contour) that is pushed from below by a “vertical” 2D half-space (or some smooth deformation of it). We establish almost optimal {C^{1,\frac{1}{2}-}} estimates for the solutions near points on the free boundary of the contact set, in any dimension {n\geq 2}. The {C^{1,\frac{1}{2}-}} estimates follow from an ε-regularity result for minimal surfaces with thin obstacles in the spirit of the De Giorgi’s improvement of flatness. To prove it, we follow Savin’s small perturbations method. A nontrivial difficulty in using Savin’s approach for minimal surfaces with thin obstacles is that near a typical contact point the solution consists of two smooth surfaces that intersect transversally, and hence it is not very flat at small scales. Via a new “dichotomy approach” based on barrier arguments we are able to overcome this difficulty and prove the desired result.