Boundary Regularity and Embedded Solutions for the Oriented Plateau Problem

Hardt, Robert; Simon, Leon

doi:10.2307/1971233

Cited by 153 publications

(235 citation statements)

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“…A very general version of this boundary regularity was proved by S. Hildebrandt; for the case of surfaces in R 3 , recall the following result of J. C. C. Nitsche. The optimal boundary regularity theorem in higher dimensions was proved by R. Hardt and L. Simon in [41].…”

Section: The Plateau Problemmentioning

confidence: 99%

Minimal Submanifolds

Colding

Minicozzi

2006

Bull. London Math. Soc.

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Section: The Plateau Problemmentioning

confidence: 99%

Minimal Submanifolds

Colding

Minicozzi

2006

Bull. London Math. Soc.

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“…Then H n−7 (singT ) ≤ c(n, α) max{c 7 2α 0 , r 1,α (∂Ω) −7 , r 1,α (S) −7 }||T ||(R n+1 ). The same proof works, using [DS02], [Bom82] in place of [HS79], [All72], and a minor modification of [NV15].…”

mentioning

confidence: 97%

“…By the maximum principle, sptT i ⊂ Ω i , and by [HS79], dist(x i , S i ) > 0. Since r 1,α (S, y) ≤ r 1,α (S, y i )/2 for y ∈ B r 1,α (S,y)/2 (y), there is no loss in generality in assuming that y i realizes the distance in S to x i .…”

mentioning

confidence: 99%

“…The problem is that there is not necessarily a good relationship between regularity and symmetry. If there exists a singular, minimizing hypersurface with Euclidean area growth and linear boundary, then by [HS79] any blow-down sequence would preclude an inclusion like singT ⊂ S n−7 ǫ (here S n−7 ǫ being the (n − 7, ǫ)-strata of [CN13]).…”

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confidence: 99%

“…Instead, for Theorem 0.1, we can prove an effective version of [HS79], which says that the singular set is some uniform distance away from the boundary curve. It's tempting to think an ineffective, quantitative version of [HS79] might hold for more general Dirichlet setups, but the problem is the same as above.…”

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confidence: 99%

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A note on the singular set of area-minimizing hypersurfaces

Edelen

2019

Calc. Var.

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We give an a priori bound on the (n − 7)-dimensional measure of the singular set for an area-minimizing n-dimensional hypersurface, in terms of the geometry of its boundary.Area-minimizing surfaces in general will not be smooth, and a basic question in minimal surface theory is to understand the size and nature of the singular region. The cumulative works of many (Federer, De Giorgi, Allard, Simons, to name only a few) prove that for absolutelyarea-minimizing n-dimensional hypersurfaces in R n+1 ("codimensionone area-minimizing integral currents"), the interior singular set is at most (n − 7)-dimensional. This dimension bound is sharp, and is directly tied to the existence of low-dimensional, non-flat minimizing cones.[HS79] proved that for such codimension-one area-minimizers, if the boundary is known to be C 1,α and multiplicity-one, then in fact no singularities lie within a neighborhood of the boundary. Combined with interior regularity, this theorem gives a very nice structure of these minimizing hypersurfaces.Recently [NV17], [NV15] quantified the interior partial regularity, by demonstrating effective local (interior) bounds on the H n−7 measure of the singular set. Their methods also prove (n − 7)-rectifiability of the singular set, which was originally established through an entirely different approach by [Sim95].In this short note, we obtain obtain a global, effective a priori estimate on the singular set of an area-minimzing hypersurface in terms of the boundary geometry. Our results are loosely analogous to the a priori bounds of [AL88] (see also the recent works [MMS18b], [MMS18a]).We work in R n+1 , for n ≥ 7. Let us write I n (U) for the space of integral n-currents acting on forms supported in the open set U. Given an n-dimensional, oriented manifold E, write [E] for the current induced by integration. Let η λ (x) = λx, and τ y (x) = x + y.If T ∈ I n (U), we say T is area-minimizing if ||T ||(W ) ≤ ||T +S||(W ) for every open W ⊂⊂ U, and every S ∈ I n (U) satisfying ∂S = 0,

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Phase transition for potentials of high‐dimensional wells

Lin

Pan

Wang

2012

Comm Pure Appl Math

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For a potential function $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}}F:\R^k \to\R _ +$ that attains its global minimum value at two disjoint compact connected submanifolds N± in $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}}\R^k$, we discuss the asymptotics, as ϵ → 0, of minimizers uϵ of the singular perturbed functional ${\bf E}_\varepsilon (u) = \int_\Omega {(|\nabla u|^2 + {1 \over {\varepsilon ^2 }}F(u))} dx$ under suitable Dirichlet boundary data $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}}g_\varepsilon :\partial \Omega \to\R ^k$. In the expansion of Eϵ (uϵ) with respect to ${1 \over \varepsilon }$, we identify the first‐order term by the area of the sharp interface between the two phases, an area‐minimizing hypersurface Γ, and the energy c 0italicF of minimal connecting orbits between N+ and N−, and the zeroth‐order term by the energy of minimizing harmonic maps into N± both under the Dirichlet boundary condition on ∂Ω and a very interesting partially constrained boundary condition on the sharp interface Γ. © 2012 Wiley Periodicals, Inc.

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Boundary Regularity and Embedded Solutions for the Oriented Plateau Problem

Cited by 153 publications

References 2 publications

Minimal Submanifolds

Minimal Submanifolds

A note on the singular set of area-minimizing hypersurfaces

Phase transition for potentials of high‐dimensional wells

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