Sliding almost minimal sets and the Plateau problem

Should We Solve Plateau’s Problem Again?

Advances in Analysis ◽

10.23943/princeton/9780691159416.003.0006 ◽

2014 ◽

Cited By ~ 1

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.

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BOUNDARY IDENTIFICATION ALGORITHMS AND VISUALIZATION OF THE SOLUTION OF THE PLATEAU PROBLEM IN A BLENDER ENVIRONMENT

Scientific Visualization ◽

10.26583/sv.9.4.02 ◽

2017 ◽

Vol 9 (4) ◽

pp. 13-25

Author(s):

E.G. Grigorieva ◽

A.A. Klyachin ◽

V.A. Klyachin

Keyword(s):

Plateau Problem ◽

Boundary Identification

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Closed Strong Spacelike Curves, Fenchel Theorem and Plateau Problem in the 3-Dimensional Minkowski Space

Chinese Annals of Mathematics Series B ◽

10.1007/s11401-019-0128-6 ◽

2019 ◽

Vol 40 (2) ◽

pp. 217-226

Author(s):

Nan Ye ◽

Xiang Ma

Keyword(s):

Minkowski Space ◽

Plateau Problem ◽

3 Dimensional ◽

Dimensional Minkowski Space

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Minimal F2-flows

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

10.1017/s1446788700022461 ◽

1985 ◽

Vol 39 (2) ◽

pp. 187-193

Author(s):

Daniel Berend

Keyword(s):

Hausdorff Dimension ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Affine Transformations ◽

Dimensional Torus ◽

Minimal Sets ◽

Necessary And Sufficient

AbstractLet σ be an ergodic endomorphism of the r–dimensional torus and Π a semigroup generated by two affine transformations lying above σ. We show that the flow defined by Π admits minimal sets of positive Hausdorff dimension and we give necessary and sufficient conditions for this flow to be minimal.

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