Prescribing the Scalar Curvature under Minimal Boundary Conditions on the Half Sphere

Ayed, Mohamed Ben; Mehdi, Khalil El; Ahmedou, Mohameden Ould

doi:10.1515/ans-2002-0201

Cited by 15 publications

(4 citation statements)

References 30 publications

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“…This problem has been studied on half spheres of dimensions n = 3, 4. See the papers [15], [12], [6], [7], [8] and the references therein. Here also in order to recover compactness one considers the following subcritical approximation (4)…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

Non simple blow ups for the Nirenberg problem on half spheres

Ahmedou¹,

Ayed²

2020

Preprint

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In this paper we study the Nirenberg problem on standard half spheres (S n + , g 0 ), which consists of finding conformal metrics of prescribed scalar curvature and zero boundary mean curvature on the boundary ∂S n + . This problem amounts to solve the following boundary value problem involving the critical Sobolev exponent:where K ∈ C 2 (S n + ) is a positive function. We construct, under generic conditions on the function K, finite energy solutions of a subcritical approximation of (P) on half spheres of dimension n ≥ 5, which exhibit multiple blow up at the same boundary point. Such a blow up phenomenon contrasts with the case of the Nirenberg problem on spheres, where non simple blow up for finite energy subsolutions cannot occur and displays an unexpected connection with vortex type problems arising in Euler equations in fluid dynamic and mean fields type equations in mathematical physics.

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Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

Non simple blow ups for the Nirenberg problem on half spheres

Ahmedou¹,

Ayed²

2020

Preprint

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“…where M is a large constant. We point out that W 22 3 is exactly the sum of of the vector fields W 3 1 (defined in (13)) with γ = −1. Furthermore, the presence of the function ψ 1 implies that the point a i moves only if |∇K 1 (a i )| ≥ M/λ i .…”

Section: Nest We Definementioning

confidence: 99%

“…This problem has been studied on half spheres of dimensions n = 2, 3, 4. See the papers [29,30,25,13,14,17,15] and the references therein. Very much like the case of spheres, to recover compactness one considers here the following subcritical approximation (4) (P ε ) −∆ g u + n(n−2)…”

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confidence: 99%

The Nirenberg problem on high dimensional half spheres: The effect of pinching conditions

Ahmedou¹,

Ayed²

2020

Preprint

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In this paper we study the Nirenberg problem on standard half spheres (S n + , g), n ≥ 5, which consists of finding conformal metrics of prescribed scalar curvature and zero boundary mean curvature on the boundary. This problem amounts to solve the following boundary value problem involving the critical Sobolev exponent:where K ∈ C 3 (S n + ) is a positive function. This problem has a variational structure but the related Euler-Lagrange functional J K lacks compactness. Indeed it admits critical points at infinity, which are limits of non compact orbits of the (negative) gradient flow. Through the construction of an appropriate pseudogradient in the neighborhood at infinity, we characterize these critical points at infinity, associate to them an index, perform a Morse type reduction of the functional J K in their neighborhood and compute their contribution to the difference of topology between the level sets of J K , hence extending the full Morse theoretical approach to this non compact variational problem. Such an approach is used to prove, under various pinching conditions, some existence results for (P) on half spheres of dimension n ≥ 5.

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“…36,39,40]. On the other hand, the case of variable functions has been discussed in some special situations, such as minimal boundaries (H = 0) [9,10,16], scalar-flat metrics (S = 0) [1,16,26,48], and various other cases involving variable curvatures [3,17,24,27].…”

Section: Introductionmentioning

confidence: 99%

Deforming convex hypersurfaces by the linear combination of the mean curvature and the square root of the scalar curvature

Sheng¹

1997

Geometry From the Pacific Rim

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On a compact manifold with boundary, the map consisting of the scalar curvature in the interior and the mean curvature on the boundary is a local surjection at generic metrics [23]. We prove that this result may be localized to compact subdomains in an arbitrary Riemannian manifold with boundary. This result is a generalization of Corvino's result [19] about localized scalar curvature deformations; however, the existence part needs to be handled delicately since the problem is non-variational. We also briefly discuss generic conditions that guarantee localized deformations, and some related geometric properties.

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Prescribing the Scalar Curvature under Minimal Boundary Conditions on the Half Sphere

Cited by 15 publications

References 30 publications

Non simple blow ups for the Nirenberg problem on half spheres

Non simple blow ups for the Nirenberg problem on half spheres

The Nirenberg problem on high dimensional half spheres: The effect of pinching conditions

Deforming convex hypersurfaces by the linear combination of the mean curvature and the square root of the scalar curvature

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