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Energy and Morse index of solutions of Yamabe type problems on thin annuli

Mohammed Ben Ayed, Khalil El Mehdi, Mohameden Ould Ahmedou, Filomena Pacella (2005)

Journal of the European Mathematical Society

We consider the Yamabe type family of problems ( P ε ) : − Δ u ε = u ε ( n + 2 ) / ( n − 2 ) , u ε > 0 in A ε , u ε = 0 on ∂ A ε , where A ε is an annulus-shaped domain of ℝ n , n ≥ 3 , which becomes thinner as ε → 0 . We show that for every solution u ε , the energy ∫ A ε | ∇ u | 2 as well as the Morse index tend to infinity as ε → 0 . This is proved through a fine blow up analysis of appropriate scalings of solutions whose limiting profiles are regular, as well as of singular solutions of some elliptic problem on ℝ n , a half-space or an infinite strip. Our argument also involves a Liouville type theorem...

Entropy of eigenfunctions of the Laplacian in dimension 2

Gabriel Rivière (2010)

Journées Équations aux dérivées partielles

We study asymptotic properties of eigenfunctions of the Laplacian on compact Riemannian surfaces of Anosov type (for instance negatively curved surfaces). More precisely, we give an answer to a question of Anantharaman and Nonnenmacher [4] by proving that the Kolmogorov-Sinai entropy of a semiclassical measure μ for the geodesic flow g t is bounded from below by half of the Ruelle upper bound. (This text has been written for the proceedings of the 37 èmes Journées EDP (Port d’Albret-June, 7-11 2010))

Estimation of vibration frequencies of linear elastic membranes

Luca Sabatini (2018)

Applications of Mathematics

The free motion of a thin elastic linear membrane is described, in a simplyfied model, by a second order linear homogeneous hyperbolic system of partial differential equations whose spatial part is the Laplace Beltrami operator acting on a Riemannian 2-dimensional manifold with boundary. We adapt the estimates of the spectrum of the Laplacian obtained in the last years by several authors for compact closed Riemannian manifolds. To make so, we use the standard technique of the doubled manifold to...

Extremal domains for the first eigenvalue of the Laplace-Beltrami operator

Frank Pacard, Pieralberto Sicbaldi (2009)

Annales de l’institut Fourier

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.

Functional inequalities and manifolds with nonnegative weighted Ricci curvature

Jing Mao (2020)

Czechoslovak Mathematical Journal

We show that n -dimensional ( n ⩾ 2 ) complete and noncompact metric measure spaces with nonnegative weighted Ricci curvature in which some Caffarelli-Kohn-Nirenberg type inequality holds are isometric to the model metric measure n -space (i.e. the Euclidean metric n -space). We also show that the Euclidean metric spaces are the only complete and noncompact metric measure spaces of nonnegative weighted Ricci curvature satisfying some prescribed Sobolev type inequality.

Generalized m-quasi-Einstein metric within the framework of Sasakian and K-contact manifolds

Amalendu Ghosh (2015)

Annales Polonici Mathematici

We consider generalized m-quasi-Einstein metric within the framework of Sasakian and K-contact manifolds. First, we prove that a complete Sasakian manifold M admitting a generalized m-quasi-Einstein metric is compact and isometric to the unit sphere S 2 n + 1 . Next, we generalize this to complete K-contact manifolds with m ≠ 1.

Currently displaying 81 – 100 of 300