Prescribing scalar and boundary mean curvature on the three dimensional half sphere

Djadli, Zindine; Malchiodi, Andrea; Ahmedou, Mohameden Ould

doi:10.1007/bf02930697

Cited by 53 publications

(62 citation statements)

References 35 publications

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“…In contrast, assumption (a) is a new feature due to the presence of the boundary and has no counterpart in the problem on all S n . (c) Theorem 6 can be the starting point to prove a global result, see [18]. Here we limit ourselves to point out that (41) can be used to evaluate the degree of I ε .…”

Section: Theorem 6 Suppose That Eithermentioning

confidence: 99%

On the Yamabe problem and the scalar curvature problems under boundary conditions

Ambrosetti¹,

Li²,

Malchiodi³

2002

Mathematische Annalen

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Section: Theorem 6 Suppose That Eithermentioning

confidence: 99%

On the Yamabe problem and the scalar curvature problems under boundary conditions

Ambrosetti¹,

Li²,

Malchiodi³

2002

Mathematische Annalen

Self Cite

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“…In the case of (P ), Ben Ayed et al proved that the balance phenomenon appears if we assume that (∂K/∂ν)(y) < 0 where y is any critical point of K 1 = K |∂S 4 + . Moreover, it is proved that this phenomenon appears also when the manifold is the three dimensional half sphere (see [15]). …”

Section: Ben Ayed R Ghoudi and K Ould Bouh Nodeamentioning

confidence: 96%

“…Li [21], and Djadli-Malchiodi-Ould Ahmedou [15] studied this problem when the manifold is the three dimensional standard half sphere. Their approach involves a fine blow up analysis of some subcritical approximations and the use of the topological degree tools.…”

Section: Ben Ayed R Ghoudi and K Ould Bouh Nodeamentioning

confidence: 99%

“…It is well known that, for the minimizing solutions, (u ε ) has to converge to a solution of (P ) or to blow up at one point which is the maximum of K or of K 1 . For the other solutions (u ε ) (not minimizing solutions) and in the case of the three dimensional half sphere, Djadli et al [15] proved that (u ε ) can blow up at (x 1 , . .…”

Section: Ben Ayed R Ghoudi and K Ould Bouh Nodeamentioning

confidence: 99%

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Existence of conformal metrics with prescribed scalar curvature on the four dimensional half sphere

Ayed

Ghoudi

Bouh

2012

Nonlinear Differ. Equ. Appl.

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Abstract. In this paper, we consider the problem of prescribing the scalar curvature under minimal boundary conditions on the four dimensional half sphere. Using dynamical and topological methods involving the study of the critical points at infinity of the associated variational structure, we prove some existence results like Bahri-Coron theorem. Furthermore, we consider the approximate subcritical problem and we construct some solutions which blow up at two different points, one of them lay on the boundary and the other one is an interior point.

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“…Related problems regarding conformal deformations of Riemannian metrics on manifolds with boundaries have been studied in [1,2,4,6,9,10,11,15,16,20,22]; see also the references therein.…”

Section: Introductionmentioning

confidence: 99%

Compactness results in conformal deformations of Riemannian metrics on manifolds with boundaries

Ahmedou¹,

Felli²

2003

Math. Z.

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Abstract. This paper is devoted to the study of a problem arising from a geometric context, namely the conformal deformation of a Riemannian metric to a scalar flat one having constant mean curvature on the boundary. By means of blow-up analysis techniques and the Positive Mass Theorem, we show that on locally conformally flat manifolds with umbilic boundary all metrics stay in a compact set with respect to the C 2 -norm and the total Leray-Schauder degree of all solutions is equal to −1 . Then we deduce from this compactness result the existence of at least one solution to our problem.MSC classification: 35J60, 53C21, 58G30.

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Prescribing scalar and boundary mean curvature on the three dimensional half sphere

Abstract: abstract. -We consider the problem of prescribing the scalar curvature and the boundary mean curvature of the standard half three sphere, by deforming conformally its standard metric. Using blow up analysis techniques and minimax arguments, we prove some existence and compactness results.

Cited by 53 publications

References 35 publications

On the Yamabe problem and the scalar curvature problems under boundary conditions

On the Yamabe problem and the scalar curvature problems under boundary conditions

Existence of conformal metrics with prescribed scalar curvature on the four dimensional half sphere

Compactness results in conformal deformations of Riemannian metrics on manifolds with boundaries

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