Concentration and confinement of eigenfunctions in a bounded open set - version 2 ; Concentration et confinement des fonctions propres dans un ouvert borné - version 2 *

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. ; Si ω et Ω, ω ⊂ Ω := (0, L) × (0, H), sont deux ouverts bornés de R 2 , il est bien connu qu'il existe une constante C ω telle que 0 < C ω ≤ R ω (u) := u 2 L 2 (ω) u 2 L 2 (Ω) < Vol(ω) Vol(Ω) pour toute fonc-tion propre u du Laplacien-Dirichlet −∆ sur Ω. Ce résultat n'étant plus exact pour un opérateur autoadjoint elliptique A du deuxième ordre sans coefficients constants, plusieurs travaux l'ont considéré dont récemment [2]. On crée une partition F N G ∪ F G de l'ensemble des fonctions propres de l'opérateur A :-les fonctions propres non guidées i.e. ∀ω = ∅, ∃C ω > 0 tel que R ω (u) > C ω , si u ∈ F N G ,-les fonctions propres guidées i.e. ∃ω = ∅, tel que inf u∈F G R ω (u) = 0. Entre autres choses, le papier caractérise ...