Regularity of minimal surfaces with lower dimensional obstacles

TL;DR

This paper proves near-optimal regularity estimates for minimal surfaces constrained by lower-dimensional obstacles, extending regularity theory to complex free boundary problems in higher dimensions.

Contribution

It introduces a new dichotomy approach and barrier arguments to establish $C^{1,1/2-}$ regularity for minimal surfaces with thin obstacles, generalizing previous results.

Findings

Established almost optimal $C^{1,1/2-}$ regularity near free boundary points.

Developed a novel dichotomy approach to handle intersecting smooth surfaces.

Extended regularity results to higher dimensions with obstacles of lower dimension.

Abstract

We 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 $C^{1,1/2-}$ estimates for the solutions near points on the free boundary of the contact set, in any dimension $n\ge 2$. The $C^{1,1/2-}$ estimates follow from an $\varepsilon$-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.