Bornes supérieures pour les valeurs propres des opérateurs naturels sur des variétés riemanniennes compactes
Hassannezhad, Asma
Colbois, Bruno
Dir.
El Soufi, Ahmad
Codir.
Ranjbar-Motlagh, Alireza
Codir.
text
thesis
Doctoral thesis
thesis
Université de Neuchâtel
Neuchâtel
eng
application/pdf
The purpose of this thesis is to find upper bounds for the eigenvalues of natural operators acting on functions on a compact Riemannian manifold (<i>M</i>, <i>g</i>) such as the Laplace-Beltrami operator and Laplace-type operators. In the case of the Laplace-Beltrami operator, two aspects are investigated: The first aspect is to study relationships between the intrinsic geometry and eigenvalues of the Laplace-Beltrami operator. In this regard, we obtain upper bounds depending only on the dimension and a conformal invariant called min-conformal volume. Asymptotically, these bounds are consistent with the Weyl law. They improve previous results by Korevaar and Yang and Yau. The proof relies on the construction of a suitable family of disjoint domains providing supports for a family of test functions. This method is powerful and interesting in itself. <br> The second aspect is to study the interplay of the extrinsic geometry and eigenvalues of the Laplace-Beltrami operator acting on compact submanifolds of <i>R</i><sup>N</sup> and of <i>CP</i><sup>N</sup>. We investigate an extrinsic invariant called the intersection index studied by Colbois, Dryden and El Soufi. For compact submanifolds of <i>R</i><sup>N</sup>, we extend their results and obtain upper bounds which are stable under small perturbation. For compact submanifolds of <i>CP</i><sup>N</sup> we obtain an upper bound depending only on the degree of submanifolds and which is sharp for the first eigenvalue. <br> As a further application of the introduced method, we obtain an upper bound for the eigenvalues of the Steklov problem in a domain with <i>C</i><sup>1</sup> boundary in a complete Riemannian manifold in terms of the isoperimetric ratio of the domain and the min-conformal volume. A modification of our method also leads to have upper bounds for the eigenvalues of Schrödinger operators in terms of the min-conformal volume and integral quantity of the potential. As another application of our method, we obtain upper bounds for the eigenvalues of the Bakry-Emery Laplace operator depending on conformal invariants and properties of the weighted function.
free
2012 Thèse de doctorat : Université de Neuchâtel, 2012
Opérateur de Laplace ; operateur de Schrödinger ; opérateur de Laplace Barky-Emery ; valeurs propres ; borne supérieure ; volume confrome minimal ; nombre d'intersection moyenne
Laplace-Beltrami operator ; Schrödinger operator ; Bakry-Emery Laplace operator ; eigenvalue ; upper bound ; min-conformal volume ; mean intersection index
Riemann, Variétés de
Schrödinger, Opérateur de
Laplacien
Valeurs propres
51
http://doc.rero.ch/record/30550/files/00002282.pdf
http://doc.rero.ch/record/30550/files/00002282.pdf
http://doc.rero.ch/record/30550
20200917135938.0
30550