Regularity of minimal surfaces with lower dimensional obstacles

2018/02/21 by Fernández-Real, Xavier, Serra, Joaquim
#35R35 #49Q05 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1802.07607

Abstract

We study the Plateau problem with a lower dimensional obstacle in ℝⁿ. Intuitively, in ℝ³ 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≥ 2. The C^1,1/2- 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.

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