On the plateau problem for surfaces of constant mean curvature

Hildebrandt, Stefan

doi:10.1002/cpa.3160230105

Cited by 118 publications

(62 citation statements)

References 8 publications

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“…In Theorems 5 and 12 below, we prove for relatively small H and boundary data, there is a solution of least energy-the small solution, and there is a large solution, with the same boundary data. This generalizes the early work of Hildebrandt [21], Brezis and Coron [6] and Struwe [29] for n = 2.…”

Section: N+1supporting

confidence: 85%

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Multiple solutions and regularity of H-systems

Mou

Yang

1996

Indiana Univ. Math. J.

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Section: N+1supporting

confidence: 85%

“…The existence of solutions and multiple solutions of (6) were established in many works, including [6] [21] [27] [29] [31] [33]. In Theorems 5 and 12 below, we prove for relatively small H and boundary data, there is a solution of least energy-the small solution, and there is a large solution, with the same boundary data.…”

Section: N+1mentioning

confidence: 94%

Multiple solutions and regularity of H-systems

Mou

Yang

1996

Indiana Univ. Math. J.

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“…== I I gki(%u%u + %v%v) du dv This answers a question posed in [2], and, in similar fashion, the conjecture in [3, §5.4] for the analytic case.…”

Section: D[x] Is the Dirichlet Integral Of X D[%]supporting

confidence: 82%

“…where the admissible X are all mappings of a domain GC.R 2 into M, continuous in G and C 2 in G, which agree with a given admissible X 0 on dG up to reparameterization ; and where H is a real constant,…”

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confidence: 99%

On regularity of generalized surfaces of constant mean curvature

Gulliver¹

1971

Bull. Amer. Math. Soc.

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Statement of result.A surface in a manifold is called "regular" to emphasize that it is immersed; the term "generalized surface" is used when regularity is only assumed almost everywhere.We treat generalized surfaces of constant mean curvature in an analytic Riemannian manifold Moi dimension 3. To this end consider the variational problem : Observe that the mapping X itself need not be a regular parameterization. Since the theorem can be applied in any neighborhood, it follows that a mapping which minimizes E even locally must be regular. The theorem is a generalization of the work of Osserman on the Plateau problem, that is, the caseConsequences. In the case that M is a space form, examples can be constructed of generalized surfaces of constant mean curvature with A MS 1970 subject classifications. Primary 49F22, 35B99, 49F25; Secondary 53B20, 49F10.

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“…We will often omit the dependence of ϕ on the parameters a, λ, R, as for δ. From (18) and some standard computations it is easy to find that, in the case R = Id…”

Section: Notation and Preliminary Factsmentioning

confidence: 99%

Asymptotic Morse Theory for the Delta-V Equation

Chanillo¹,

Malchiodi²

2005

Communications in Analysis and Geometry

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Given a smooth bounded domain Ω ⊆ R 2 , we consider the equation ∆v = 2v x ∧ v y in Ω, where v : Ω → R 3 . We prescribe Dirichlet boundary datum, and consider the case in which this datum converges to zero. An asymptotic study of the corresponding Euler functional is performed, analyzing multiple-bubbling phenomena. This allows us to settle a particular case of a question raised by H. Brezis and J.M. Coron in [9].

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On the plateau problem for surfaces of constant mean curvature

Cited by 118 publications

References 8 publications

Multiple solutions and regularity of H-systems

Multiple solutions and regularity of H-systems

On regularity of generalized surfaces of constant mean curvature

Asymptotic Morse Theory for the Delta-V Equation

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