scholarly journals Free boundary minimal surfaces: a nonlocal approach

2020 ◽  
Vol 20 (2) ◽  
pp. 437-489 ◽  
Author(s):  
Francesca Da Lio ◽  
Alessandro Pigati
Keyword(s):  
Free Boundary ◽  
Minimal Surfaces ◽  
Nonlocal Approach

scholarly journals Free boundary minimal surfaces in the unit ball with low cohomogeneity

2016 ◽  
Vol 145 (4) ◽  
pp. 1671-1683 ◽  
Author(s):  
Brian Freidin ◽  
Mamikon Gulian ◽  
Peter McGrath
Keyword(s):  
Free Boundary ◽  
Unit Ball ◽  
Minimal Surfaces

Approximation of minimal surfaces with free boundaries: convergence results

2018 ◽  
Vol 39 (3) ◽  
pp. 1391-1420
Author(s):  
Tristan Jenschke
Keyword(s):  
Free Boundary ◽  
Boundary Problem ◽  
Free Boundary Problem ◽  
Minimal Surfaces ◽  
Approximate Solutions ◽  
Variational Problems ◽  
Free Boundaries ◽  
Convergence Estimate ◽  
Convergence Results ◽  
Discrete Finite Element

Abstract In a previous paper we developed a penalty method to approximate solutions of the free boundary problem for minimal surfaces by solutions of certain variational problems depending on a parameter $\lambda $. There we showed existence and $C^2$-regularity of these solutions as well as convergence to the solution of the free boundary problem for $\lambda \to \infty $. In this paper we develop a fully discrete finite element procedure for approximating solutions of these variational problems and prove a convergence estimate, which includes an order of convergence with respect to the grid size.

Optimal boundary regularity for minimal surfaces with a free boundary

1981 ◽  
Vol 33 (3-4) ◽  
pp. 357-364 ◽  
Author(s):  
Stefan Hildebrandt ◽  
Johannes C. C. Nitsche
Keyword(s):  
Free Boundary ◽  
Minimal Surfaces ◽  
Boundary Regularity ◽  
Optimal Boundary

scholarly journals Free boundary minimal surfaces in the unit 3-ball

2017 ◽  
Vol 154 (3-4) ◽  
pp. 359-409 ◽  
Author(s):  
Abigail Folha ◽  
Frank Pacard ◽  
Tatiana Zolotareva
Keyword(s):  
Free Boundary ◽  
Minimal Surfaces

scholarly journals Sharp Area Bounds for Free Boundary Minimal Surfaces in Conformally Euclidean Balls

2018 ◽  
Vol 2020 (18) ◽  
pp. 5630-5641 ◽  
Author(s):  
Brian Freidin ◽  
Peter McGrath
Keyword(s):  
Free Boundary ◽  
Minimal Surface ◽  
Minimal Surfaces ◽  
Geodesic Ball ◽  
Euclidean Setting

Abstract We prove that the area of a free boundary minimal surface $\Sigma ^2 \subset B^n$, where $B^n$ is a geodesic ball contained in a round hemisphere $\mathbb{S}^n_+$, is at least as big as that of a geodesic disk with the same radius as $B^n$; equality is attained only if $\Sigma $ coincides with such a disk. More generally, we prove analogous results for a class of conformally euclidean ambient spaces. This follows works of Brendle and Fraser–Schoen in the euclidean setting.

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