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Regularity of minimal surfaces with lower-dimensional obstacles

Xavier Fernández-Real and Joaquim Serra

Abstract

We study the Plateau problem with a lower-dimensional obstacle in n . Intuitively, in 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 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.

Funding statement: This work has received funding from the European Research Council (ERC) under the Grant Agreement No. 721675. The second author was also supported by the Swiss National Science Foundation (Ambizione grant PZ00P2_180042).

Acknowledgements

We thank M. Focardi, G. De Philippis, and E. Spadaro, for interesting discussions on the topic of the paper. We would also like to deeply thank Connor Mooney for pointing out to us the key observation that yields the classification of minimal cones in every dimension (see Proposition 1.9).

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Received: 2018-10-10
Revised: 2019-07-17
Published Online: 2019-11-09
Published in Print: 2020-10-01

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