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Sicbaldi Pieralberto - One of the best experts on this subject based on the ideXlab platform.
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New examples of extremal domains for the first eigenvalue of the Laplace-Beltrami operator in a Riemannian manifold with boundary
'Oxford University Press (OUP)', 2016Co-Authors: Lamboley Jimmy, Sicbaldi PieralbertoAbstract:We build new examples of extremal domains with small prescribed volume for the first eigenvalue ofthe Laplace-Beltrami operator in some Riemannian manifold with boundary. These domains are close to half ballsof small radius centered at a Nondegenerate Critical Point of the mean curvature function of the boundary of themanifold, and their boundary intersects the boundary of the manifold orthogonally.nonnonouirechercheInternationa
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Extremal domains for the first eigenvalue in a general compact Riemannian manifold
'American Institute of Mathematical Sciences (AIMS)', 2015Co-Authors: Delay Erwann, Sicbaldi PieralbertoAbstract:International audienceWe prove the existence of extremal domains with small prescribed volume for the first eigenvalue of the Laplace-Beltrami operator in any compact Riemannian manifold. This result generalizes a results of F. Pacard and the second author where the existence of a Nondegenerate Critical Point of the scalar curvature of the Riemannian manifold was required
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Extremal domains of big volume for the first eigenvalue of the Laplace-Beltrami operator in a compact manifold
'Elsevier BV', 2014Co-Authors: Sicbaldi PieralbertoAbstract:International audienceWe prove the existence of extremal domains for the first eigenvalue of the Laplace-Beltrami operator in some compact Riemannian manifolds, with volume close to the volume of the manifold. If the first (positive) eigenfunction F of the Laplace-Beltrami operator over the manifold is a nonconstant function, these domains are close to the complement of geodesic balls of small radius whose center is close to the Point where F attains its maximum. If F is a constant function and the dimension of the manifold is at least 4, these domains are close to the complement of geodesic balls of small radius whose center is close to a Nondegenerate Critical Point of the scalar curvature function
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New examples of extremal domains for the first eigenvalue of the Laplace-Beltrami operator in a Riemannian manifold with boundary
2014Co-Authors: Lamboley Jimmy, Sicbaldi PieralbertoAbstract:We build new examples of extremal domains with small prescribed volume for the first eigenvalue of the Laplace-Beltrami operator in some Riemannian manifold with boundary. These domains are close to half balls of small radius centered at a Nondegenerate Critical Point of the mean curvature function of the boundary of the manifold, and their boundary intersects the boundary of the manifold orthogonally.Comment: 30 pages, 3 figure
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Extremal domains for the first eigenvalue in a general Riemannian manifold
2013Co-Authors: Delay Erwann, Sicbaldi PieralbertoAbstract:We prove the existence of extremal domains with small prescribed volume for the first eigenvalue of the Laplace-Beltrami operator in any compact Riemannian manifold. This result generalizes a results of F. Pacard and the second author where the existence of a Nondegenerate Critical Point of the scalar curvature of the Riemannian manifold was required.Comment: 29 page
Pacard Frank - One of the best experts on this subject based on the ideXlab platform.
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Extremal domains for the first eigenvalue of the Laplace-Beltrami operator
'Cellule MathDoc CEDRAM', 2009Co-Authors: Pacard Frank, Sicbaldi PieralbertoAbstract:International audienceWe prove the existence of extremal domains with small prescribed volume for the first eigenvalue of Laplace-Beltrami operator in some Riemannian manifold. These domains are close to geodesic spheres of small radius centered at a Nondegenerate Critical Point of the scalar curvature
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Domaines extrémaux pour la première valeur propre de l'opérateur de Laplace-Beltrami
2009Co-Authors: Sicbaldi Pieralberto, Pacard FrankAbstract:Dans tout ce qui suit, nous considérons une variété riemannienne compacte de dimension au moins égale à 2. A tout domaine (suffisamment régulier) , on peut associer la première valeur propre ?Ù de l opérateur de Laplace-Beltrami avec condition de Dirichlet au bord. Nous dirons qu un domaine est extrémal (sous entendu, pour la première valeur propre de l opérateur de Laplace-Beltrami) si est un Point critique de la fonctionnelle Ù? ?O sous une contrainte de volume V ol(Ù) = c0. Autrement dit, est extrémal si, pour toute famille régulière {Ot}te (-t0,t0) de domaines de volume constant, telle que Ù 0 = Ù, la dérivée de la fonction t ? ?Ot en 0 est nulle. Rappelons que les domaines extrémaux sont caractérisés par le fait que la fonction propre, associée à la première valeur propre sur le domaine avec condition de Dirichlet au bord, a une donnée de Neumann constante au bord. Ce résultat a été démontré par A. El Soufi et S. Ilias en 2007. Les domaines extrémaux sont donc des domaines sur lesquels peut être résolu un problème elliptique surdéterminé. L objectif principal de cette thèse est la construction de domaines extrémaux pour la première valeur propre de l opérateur de Laplace-Beltrami avec condition de Dirichlet au bord. Nous donnons des résultats d existence de domaines extrémaux dans le cas de petits volumes ou bien dans le cas de volumes proches du volume de la variété. Nos résultats permettent ainsi de donner de nouveaux exemples non triviaux de domaines extrémaux. Le premier résultat que nous avons obtenu affirme que si une variété admet un Point critique non dégénéré de la courbure scalaire, alors pour tout volume petit il existe un domaine extrémal qui peut être construit en perturbant une boule géodésique centrée en ce Point critique non dégénéré de la courbure scalaire. La méthode que nous utilisons pour construire ces domaines extrémaux revient à étudier l opérateur (non linéaire) qui à un domaine associe la donnée de Neumann de la première fonction propre de l opérateur de Laplace-Beltrami sur le domaine. Il s agit d un opérateur (hautement non linéaire), nonlocal, elliptique d ordre 1. Dans Rn . R/Z, le domaine cylindrique Br . R/Z, ou Br est la boule de rayon r > 0 dans Rn, est un domaine extrémal. En étudiant le linéarisé de l opérateur elliptique du premier ordre défini par le problème précédent et en utilisant un résultat de bifurcation, nous avons démontré l existence de domaines extrémaux nontriviaux dans Rn . R/Z. Ces nouveaux domaines extrémaux sont proches de domaines cylindriques Br . R/Z. S ils sont invariants par rotation autour de l axe vertical, ces domaines ne sont plus invariants par translations verticales. Ce deuxieme r esultat donne un contre-exemple à une conjecture de Berestycki, Caffarelli et Nirenberg énoncée en 1997. Pour de grands volumes la construction de domaines extrémaux est techniquement plus difficile et fait apparaître des phénomènes nouveaux. Dans ce cadre, nous avons dû distinguer deux cas selon que la première fonction propre Ø0 de l opérateur de Laplace-Beltrami sur la variété est constante ou non. Les résultats que nous avons obtenus sont les suivants : 1. Si Ø0 a des Points critiques non dégénérés (donc en particulier n est pas constante), alors pour tout volume assez proche du volume de la variété, il existe un domaine extrémal obtenu en perturbant le complément d une boule géodésique centrée en un des Points critiques non dégénérés de Ø0. 2. Si Ø0 est constante et la variété admet des Points critiques non dégénérés de la courbure scalaire, alors pour tout volume assez proche du volume de la variété il existe un domaine extrémal obtenu en perturbant le complément d une boule géodésique centrée en un des Points critiques non dégénérés de la courbure scalaireIn what follows, we will consider a compact Riemannian manifold whose dimension is at least 2. Let Ù be a (smooth enough) domain and ?O the first eigenvalue of the Laplace-Beltrami operator on Ù with 0 Dirichlet boundary condition. We say that Ù is extremal (for the first eigenvalue of the Laplace-Beltrami operator) if is a Critical Point for the functional Ù? ?O with respect to variations of the domain which preserve its volume. In other words, Ù is extremal if, for all smooth family of domains { Ù t}te(-t0,t0) whose volume is equal to a constant c0, and Ù 0 = Ù, the derivative of the function t ? ?Ot computed at t = 0 is equal to 0. We recall that an extremal domain is characterized by the fact that the eigenfunction associated to the first eigenvalue of the Laplace-Beltrami operator over the domain with 0 Dirichlet boundary condition, has constant Neumann data at the boundary. This result has been proved by A. El Soufi and S. Ilias in 2007. Extremal domains are then domains over which can be solved an elliptic overdeterminated problem. The main aim of this thesis is the construction of extremal domains for the first eigenvalue of the Laplace-Beltrami operator with 0 Dirichlet boundary condition. We give some existence results of extremal domains in the cases of small volume or volume closed to the volume of the manifold. Our results allow also to construct some new nontrivial exemples of extremal domains. The first result we obtained states that if the manifold has a Nondegenerate Critical Point of the scalar curvature, then, given a fixed volume small enough, there exists an extremal domain that can be constructed by perturbation of a geodesic ball centered in that Nondegenerated Critical Point of the scalar curvature. The methode used is based on the study of the operator that to a given domain associes the Neumann data of the first eigenfunction of the Laplace-Beltrami operator over the domain. It is a highly nonlinear, non local, elliptic first order operator. In Rn . R/Z, the circular-cylinder-type domain Br . R/Z, where Br is the ball of radius r > 0 in Rn, is an extremal domain. By studying the linearized of the elliptic first order operator defined in the previous problem, and using some bifurcation results, we prove the existence of nontrivial extremal domains in Rn . R/Z. Such extremal domains are closed to the circular-cylinder-type domains Br . R/Z. If they are invariant by rotation with respect to the vertical axe, they are not invariant by vertical translations. This second result gives a counterexemple to a conjecture of Berestycki, Caffarelli and Nirenberg stated in 1997. For big volumes the construction of extremal domains is technically more difficult and shows some new phenomena. In this context, we had to distinguish two cases, according to the fact that the first eigenfunction Ø0 of the Laplace-Beltrami operator over the manifold is constant or not. The results obtained are the following : 1. If Ø0 has a Nondegenerated Critical Point (in particular it is not constant), then, given a fixed volume closed to the volume of the manifold, there exists an extremal domain obtained by perturbation of the complement of a geodesic ball centered in a Nondegenerated Critical Point of Ø0. 2. If Ø0 is constant and the manifold has some Nondegenerate Critical Points of the scalar curvature, then, for a given fixed volume closed to the volume of the manifold, there exists an extremal domain obtained by perturbation of the complement of a geodesic ball centered in a Nondegenerate Critical Point of the scalar curvaturePARIS-EST-Université (770839901) / SudocSudocFranceF
Eduard Zehnder - One of the best experts on this subject based on the ideXlab platform.
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morse theory for periodic solutions of hamiltonian systems and the maslov index
Communications on Pure and Applied Mathematics, 1992Co-Authors: Dietmar Salamon, Eduard ZehnderAbstract:In this paper we prove Morse type inequalities for the contractible 1-periodic solutions of time dependent Hamiltonian differential equations on those compact symplectic manifolds M for which the symplectic form and the first Chern class of the tangent bundle vanish over q(M). The proof is based on a version of infinite dimensional Morse theory which is due to Floer. The key Point is an index theorem for the Fredholm operator which plays a central role in Floer homology. The index formula involves the Maslov index of Nondegenerate contractible periodic solutions. This Maslov index plays the same role as the Morse index of a Nondegenerate Critical Point does in finite dimensional Morse theory. We shall use this connection between Floer homology and Maslov index to establish the existence of infinitely many periodic solutions having integer periods provided that every I-periodic solution has at least one Floquet multiplier which is not equal to 1.
Khalil El Mehdi - One of the best experts on this subject based on the ideXlab platform.
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Single blow-up solutions for a slightly subCritical biharmonic equation
Hindawi Limited, 2006Co-Authors: Khalil El MehdiAbstract:We consider a biharmonic equation under the Navier boundary condition and with a nearly Critical exponent (Pε): ∆2u=u9−ε, u>0 in Ω and u=∆u=0 on ∂Ω, where Ω is a smooth bounded domain in ℝ5, ε>0. We study the asymptotic behavior of solutions of (Pε) which are minimizing for the Sobolev quotient as ε goes to zero. We show that such solutions concentrate around a Point x0∈Ω as ε→0, moreover x0 is a Critical Point of the Robin's function. Conversely, we show that for any Nondegenerate Critical Point x0 of the Robin's function, there exist solutions of (Pε) concentrating around x0 as ε→0
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SINGLE BLOW-UP SOLUTIONS FOR A SLIGHTLY SUBCritical BIHARMONIC EQUATION
2005Co-Authors: Khalil El MehdiAbstract:We consider a biharmonic equation under the Navier boundary condition and with a nearly Critical exponent (Pε): Δ2u = u9−ε, u> 0 in Ω and u = Δu = 0 on ∂Ω, where Ω is a smooth bounded domain in R5, ε> 0. We study the asymptotic behavior of solutions of (Pε) which are minimizing for the Sobolev quotient as ε goes to zero. We show that such solutions concentrate around a Point x0 ∈Ω as ε → 0, moreover x0 is a Critical Point of the Robin’s function. Conversely, we show that for any Nondegenerate Critical Point x0 of the Robin’s function, there exist solutions of (Pε) concentrating around x0 as ε → 0. Copyright © 2006 Khalil El Mehdi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited
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On a Biharmonic Equation Involving Nearly Critical Exponent
2004Co-Authors: Ayed, Mohamed Ben, Khalil El MehdiAbstract:This paper is concerned with a biharmonic equation under the Navier boundary condition with nearly Critical exponent. We study the asymptotic behavior os solutions which are minimizing for the Sobolev quatient. We show that such solutions concentrate around an interior Point which is a Critical Point of the Robin's function. Conversely, we show that for any Nondegenerate Critical Point fo the Robin's function, the exist solutions concentrating around such a Point. Finally, we prove that, in contrast with what happened in the subCritical equation, the superCritical problem has no solutions which concentrate around a Point .Comment: 23 page
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Single Blow-up Solutions for a Slightly SubCritical Biharmonic Equation
2004Co-Authors: Khalil El MehdiAbstract:Abstract. In this paper, we consider a biharmonic equation under the Navier boundary condition and with a nearly Critical exponent (Pε): ∆ 2 u = u 9−ε, u> 0 in Ω and u = ∆u = 0 on ∂Ω, where Ω is a smooth bounded domain in R 5, ε> 0. We study the asymptotic behavior of solutions of (Pε) which are minimizing for the Sobolev quotient as ε goes to zero. We show that such solutions concentrate around a Point x0 ∈ Ω as ε → 0, moreover x0 is a Critical Point of the Robin’s function. Conversely, we show that for any Nondegenerate Critical Point x0 of the Robin’s function, there exist solutions of (Pε) concentrating around x0 as ε → 0
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Single Blow up Solutions for a Slightly SubCritical Biharmonic Equation
2004Co-Authors: Khalil El MehdiAbstract:In this paper, we consider a biharmonic equation under the Navier boundary condition and with a nearly Critical exponent $(P_\epsilon): \Delta^2u=u^{9-\epsilon}, u>0$ in $\Omega$ and $u=\Delta u=0$ on $\partial\Omega$, where $\Omega$ is a smooth bounded domain in $\R^5$ and $\epsilon >0$. We study the asymptotic behavior of solutions of $(P_\epsilon)$ which are minimizing for the Sobolev qutient as $\epsilon$ goes to zero. We show that such solutions concentrate around a Point $x_0\in\Omega$ as $\epsilon\to 0$, moreover $x_0$ is a Critical Point of the Robin's function. Conversely, we show that for any Nondegenerate Critical Point $x_0$ of the Robin's function, there exist solutions concentrating around $x_0$ as $\epsilon$ goes to zero.Comment: 19 page
Pieralberto Sicbaldi - One of the best experts on this subject based on the ideXlab platform.
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Extremal domains for the first eigenvalue in a general compact Riemannian manifold
Discrete and Continuous Dynamical Systems, 2015Co-Authors: Erwann Delay, Pieralberto SicbaldiAbstract:We prove the existence of extremal domains with small prescribed volume for the first eigenvalue of the Laplace-Beltrami operator in any compact Riemannian manifold. This result generalizes a results of F. Pacard and the second author where the existence of a Nondegenerate Critical Point of the scalar curvature of the Riemannian manifold was required.
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Contents
2013Co-Authors: Erwann Delay, Pieralberto SicbaldiAbstract:Abstract. We prove the existence of extremal domains with small prescribed volume for the first eigenvalue of the Laplace-Beltrami operator in any compact Riemannian manifold. This result generalizes a results of F. Pacard and the second author where the existence of a Nondegenerate Critical Point of the scalar curvature of the Riemannian manifold was required
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Extremal domains for the first eigenvalue of the Laplace-Beltrami operator
Annales de l’institut Fourier, 2009Co-Authors: Frank Pacard, Pieralberto SicbaldiAbstract:We prove the existence of extremal domains with small prescribed volume for the first eigenvalue of Laplace-Beltrami operator in some Riemannian manifold. These domains are close to geodesic spheres of small radius centered at a Nondegenerate Critical Point of the scalar curvature.