scholarly journals Should We Solve Plateau’s Problem Again?

2014 ◽  
Author(s):  
Guy David
Keyword(s):  
Local Regularity ◽  
Plateau Problem ◽  
Soap Films ◽  
Plateau’S Problem ◽  
Minimization Problems ◽  
Minimal Sets ◽  
Modeling Problem ◽  
Regularity Properties ◽  
Plateau's Problem ◽  
Regularity Results

This chapter gives a partial account of the situation of Plateau's problem on the existence and regularity of soap films with a given boundary. It starts with a description of some of the most celebrated solutions of Plateau's problem, followed by a description of a few easy examples. The chapter then returns to the modeling problem and mentions a few additional ways to state a Plateau problem. It briefly describes the known local regularity properties of the Almgren minimal sets, and why we would like to extend some of these regularity results to sliding minimal sets, all the way to the boundary. At the same time, the chapter considers why these solutions are not always entirely satisfactory. Finally, the chapter explains why the regularity results for sliding Almgren minimal sets also apply to solutions of the Reifenberg and size minimization problems described earlier in the chapter.

scholarly journals Hölder regularity at the boundary of two-dimensional sliding almost minimal sets

2018 ◽  
Vol 11 (1) ◽  
pp. 29-63
Author(s):  
Yangqin Fang
Keyword(s):  
Regularity Result ◽  
Hölder Regularity ◽  
Two Dimensional ◽  
Soap Films ◽  
Regularity Theorem ◽  
Plateau’S Problem ◽  
Minimal Sets ◽  
Plateau's Problem ◽  
Dimension 2 ◽  
Holder Regularity

AbstractIn [15], Jean Taylor proved a regularity theorem away from the boundary for Almgren almost minimal sets of dimension 2 in {\mathbb{R}^{3}}. It is quite important for understanding the soap films and the solutions of Plateau’s problem away from boundary. In this paper, we will give a regularity result on the boundary for two-dimensional sliding almost minimal sets in {\mathbb{R}^{3}}.

scholarly journals On solving the plateau problem in parametric form

1983 ◽  
Vol 6 (2) ◽  
pp. 341-361
Author(s):  
Baruch cahlon ◽  
Alan D. Solomon ◽  
Louis J. Nachman
Keyword(s):  
Minimal Surface ◽  
Numerical Method ◽  
Minimal Surfaces ◽  
Plateau Problem ◽  
Parametric Form ◽  
First Variation ◽  
Plateau’S Problem ◽  
Numerical Example ◽  
Plateau's Problem ◽  
The First Variation

This paper presents a numerical method for finding the solution of Plateau's problem in parametric form. Using the properties of minimal surfaces we succeded in transferring the problem of finding the minimal surface to a problem of minimizing a functional over a class of scalar functions. A numerical method of minimizing a functional using the first variation is presented and convergence is proven. A numerical example is given.

scholarly journals Covers, soap films and BV functions

2018 ◽  
Vol 3 (1) ◽  
pp. 57-75 ◽  
Author(s):  
Giovanni Bellettini ◽  
Maurizio Paolini ◽  
Franco Pasquarelli ◽  
Giuseppe Scianna
Keyword(s):  
Soap Film ◽  
Space Curve ◽  
Soap Films ◽  
Plateau’S Problem ◽  
Bv Functions ◽  
Double Covers ◽  
Plateau's Problem ◽  
Large Tunnel

Abstract In this paper we review the double covers method with constrained BV functions for solving the classical Plateau’s problem. Next, we carefully analyze some interesting examples of soap films compatible with covers of degree larger than two: in particular, the case of a soap film only partially wetting a space curve, a soap film spanning a cubical frame but having a large tunnel, a soap film that retracts to its boundary, and various soap films spanning an octahedral frame.

scholarly journals Local Regularity for the Harmonic Map and Yang–Mills Heat Flows

2021 ◽  
Author(s):  
Ahmad Afuni
Keyword(s):  
Riemannian Manifolds ◽  
Harmonic Map ◽  
Local Regularity ◽  
Yang Mills ◽  
Heat Flows ◽  
Singular Sets ◽  
Regularity Results ◽  
Local Monotonicity ◽  
Partial Differ ◽  
Singular Time

AbstractWe establish new local regularity results for the harmonic map and Yang–Mills heat flows on Riemannian manifolds of dimension greater than 2 and 4, respectively, obtaining criteria for the smooth local extensibility of these flows. As a corollary, we obtain new characterisations of singularity formation and use this to obtain a local estimate on the Hausdorff measure of the singular sets of these flows at the first singular time. Finally, we show that smooth blow-ups at rapidly forming singularities of these flows are necessarily nontrivial and admit a positive lower bound on their heat ball energies. These results crucially depend on some local monotonicity formulæ for these flows recently established by Ecker (Calc Var Partial Differ Equ 23(1):67–81, 2005) and the Afuni (Calc Var 555(1):1–14, 2016; Adv Calc Var 12(2):135–156, 2019).

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