The Dirichlet problem for the minimal hypersurface equation on arbitrary domains of a Riemannian manifold

Ari, Aiolfi,; Ripoll, Jaime; Soret, Marc

doi:10.1007/s00229-015-0774-2

Cited by 9 publications

(16 citation statements)

References 30 publications

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“…This is proved in [12] using classical Plateau's problem technique. (c) The EDP for the minimal surface equation is studied in the Riemannian setting in [1] and [3].…”

Section: Remarksmentioning

confidence: 99%

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On the existence of foliations by solutions to the exterior Dirichlet problem for the minimal surface equation

Ari¹,

Bustos²,

Ripoll³

2020

Preprint

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Given an exterior domain Ω of C 2,α class of R n , n ≥ 3, it is proved the existence of a foliation of the closure of an open set of R n+1 \ (∂Ω × R) which leaves are graphs of the solutions of the exterior Dirichlet problem on Ω for the minimal surface equation containing the trivial solution. The leaves have horizontal ends and are parametrized by the maximum angle α ∈ [0, π/2] that the Gauss map in R n+1 of the leaves make with the vertical direction at the boundary of the domain. The solutions have a limit c α at infinity, the map α → c α is increasing and bounded. Moreover, if R n \Ω satisfies the interior sphere condition of maximal radius ρ and if ∂Ω is contained in a ball of minimal radius ̺ thenwhereOne of the above inclusions is an equality if and only if ρ = ̺, Ω is the exterior of a ball of radius ρ and the solutions are radial. These foliations were studied by E. Kuwert in [6] and our result answers a natural question about the existence of such foliations which was not touched in [6].

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“…This is proved in [12] using classical Plateau's problem technique. (c) The EDP for the minimal surface equation is studied in the Riemannian setting in [1] and [3].…”

Section: Remarksmentioning

confidence: 99%

“…where Ω ⊂ R n , n ≥ 2, is an exterior domain that is, Λ := R n \Ω is a relatively compact domain, and ϕ ∈ C 0 (∂Ω) a given function. Additionally to existence or not of solutions of (1), one is also interested on global properties of their graphs in R n+1 .…”

Section: Introductionmentioning

confidence: 99%

On the existence of foliations by solutions to the exterior Dirichlet problem for the minimal surface equation

Ari¹,

Bustos²,

Ripoll³

2020

Preprint

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“…Theorem 2 (Sharp Serrin-type solvability criterion). Let M be a simply connected and compact manifold whose sectional curvature satisfies 1 4…”

Section: P 787]mentioning

confidence: 99%

“…It has been proved in chronological order by Finn [8], Jenkins-Serrin [13] and Serrin [17], that the very well known Serrin condition is a necessary condition for the solvability of the Dirichlet problem for equation (1) in bounded domains of R n .…”

Section: Introductionmentioning

confidence: 99%

“…Dirichlet problems for equations whose solutions describe hypersurfaces of prescribed mean curvature have been studied in manifolds different from the Euclidean space. Several works have considered Serrin type conditions that provide some existence theorems (see [1], [2], [6], [7], [12], [15], [16] and [18] as examples). However, non-existence theorems have been only investigated in a few cases that we summarize below.…”

Section: Introductionmentioning

confidence: 99%

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Sharp solvability criteria for Dirichlet problems of mean curvature type in Riemannian manifolds: non-existence results

Earp

2019

Calc. Var.

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It is well known that the Serrin condition is a necessary condition for the solvability of the Dirichlet problem for the prescribed mean curvature equation in bounded domains of R n with certain regularity. In this paper we investigate the sharpness of the Serrin condition for the vertical mean curvature equation in the product M n × R. Precisely, given a C 2 bounded domain Ω in M and a function H = H(x, z) continuous in Ω × R and nondecreasing in the variable z, we prove that the strong Serrin condition (n − 1)H ∂Ω (y) ≥ n sup z∈R |H(y, z)| ∀ y ∈ ∂Ω, is a necessary condition for the solvability of the Dirichlet problem in a large class of Riemannian manifolds within which are the Hadamard manifolds and manifolds whose sectional curvatures are bounded above by a positive constant. As a consequence of our results we deduce Jenkins-Serrin and Serrin type sharp solvability criteria. * Supported by CAPES and CNPq of Brazil. † Partially supported by CNPq of Brazil. 2000 AMS Subject Classification: 53C42, 49Q05, 35J25, 35J60.

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Compact CMC Graphs in $$M\times \mathbb {R}$$ M × R with Boundary in Two Horizontal Slices

Ari

Nunes

Sauer

et al. 2017

Bull Braz Math Soc, New Series

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The Dirichlet problem for the minimal hypersurface equation on arbitrary domains of a Riemannian manifold

Abstract: International audienc

Cited by 9 publications

References 30 publications

On the existence of foliations by solutions to the exterior Dirichlet problem for the minimal surface equation

On the existence of foliations by solutions to the exterior Dirichlet problem for the minimal surface equation

Sharp solvability criteria for Dirichlet problems of mean curvature type in Riemannian manifolds: non-existence results

Compact CMC Graphs in $$M\times \mathbb {R}$$ M × R with Boundary in Two Horizontal Slices

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