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Concentration et confinement des fonctions propres dans un ouvert borné - version 2 *

Abstract : Let −∆ be the Laplacian in Ω := (0, L) × (0, H), subject to Dirichlet boundary conditions and let u be an eigenfunction of −∆. For any open set ω ⊂ Ω define R ω (u) = u 2 L 2 (ω) u 2 L 2 (Ω). It is well known that there exists a constant C ω > 0 such that C ω ≤ R ω (u) for all eigenfunctions. This is no longer true for certain more general second-order elliptic operators and many authors have considered this subject whose [2] recently. This work is concerned with such operators, occuring in "layered media". In this more general case the set of eigenfunctions is the disjoint union of two non-empty sets F N G ∪ F G as follows.-non-guided eigenfunctions : ∀ω = ∅, any u ∈ F N G satisfies R ω (u) > C ω , * fichier : concentration-confinement-reformatage-v2tertio-ter.tex 1-guided eigenfunctions : ∃ω, ω = ∅, such that inf u∈F G R ω (u) = 0. The paper deals with a spectral characterization of theses two sets among others things. The layered structure of the operator permits a representation of its spectrum as a subset of points indexed by (k, l) ∈ N × N. This allows a geometric description of the guided and non-guided eigenfunction categories. Section 4.1 of the previous version was not correct, now it is corrected, many proofs are simplified and a new general result is added.
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Contributor : Yves Dermenjian <>
Submitted on : Monday, August 31, 2020 - 4:37:28 PM
Last modification on : Wednesday, September 2, 2020 - 3:31:51 AM


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  • HAL Id : hal-02926376, version 1



Assia Benabdallah, Matania Ben-Artzi, Yves Dermenjian. Concentration et confinement des fonctions propres dans un ouvert borné - version 2 *. 2020. ⟨hal-02926376⟩



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