Regularity of minimal surfaces with lower-dimensional obstacles
OPEN ACCESS
Author / Producer
Date
2020-10
Publication Type
Journal Article
ETH Bibliography
yes
Citations
Altmetric
OPEN ACCESS
Data
Rights / License
Abstract
We study the Plateau problem with a lower-dimensional obstacle in R-n. Intuitively, in 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,C- 1/2- estimates for the solutions near points on the free boundary of the contact set, in any dimension n >= 2. The C-1,C-1/2- estimates follow from an epsilon-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.
Permanent link
Publication status
published
External links
Editor
Book title
Journal / series
Volume
767
Pages / Article No.
37 - 75
Publisher
De Gruyter
Event
Edition / version
Methods
Software
Geographic location
Date collected
Date created
Subject
Organisational unit
Notes
Funding
721675 - Regularity and Stability in Partial Differential Equations (EC)
180042 - A geometric approach to nonlinear elliptic and parabolic equations (SNF)
180042 - A geometric approach to nonlinear elliptic and parabolic equations (SNF)