CONCENTRATION AND NON-CONCENTRATION OF EIGENFUNCTIONS OF SECOND-ORDER ELLIPTIC OPERATORS WITH A DIVERGENCE FORM IN LAYERED MEDIA
Concentration et non-concentration de fonctions propres d'opérateurs elliptiques divergentiels d'ordre 2 en milieux stratifiés
Résumé
Let $Ω ′ ⊂ R^d , d = 1, 2, . . . $ be an open bounded smooth domain, and
$$\Omega = \Omega'\times (0,H)\subset \mathbb{R}^d \times \mathbb_+.$$ The coordinates in Ω are designated as x = (x ′ , y) ∈ Ω ′ × (0, H). The paper deals with the concentration (and non-concentration) properties (in sectors of Ω) of the eigenfunctions of the self-adjoint second-order elliptic operator
$A = -\nabla\cdot\tilde{c}\nabla$ in $L^2(\Omega,dx)$ with domain $D(A) = \{v\in H_0^1(\Omega); \tilde{c}\nabla v \in H^1(\Omega)\}.$
The coefficient $\tilde{c}>0$ is assumed to be bounded, but no continuity assumption is imposed. It is analogous to the square of the speed of sound in the wave equation, and $\square{\tilde{c}}$ is commonly known in the physical literature as the celerity. This study deals with layered media, namely, $\tilde{c}(x)$) depends only on the single spatial coordinate y ∈ (0, H), so that $\tilde{c}(x) = \tilde{c}(x ′ , y) = c(y).$ The eigenvalues of A are partitioned (apart from a small residual set) into two disjoint infinite sets. The corresponding eigenfunctions are labeled as $F_G$ (guided) and $F_{N G}$ (non-guided). Their asymptotic properties are expressed by suitable estimates as the associated eigenvalues tend to infinity. The eigenfunctions in $F_ G$ concentrate in "wells" of $c(y),$ subject to polynomial rate of decay away from the concentration sector. The non-concentrating eigenfunctions in $F_{N G}$ are oscillatory in every sector with non-decaying amplitudes. These results hold uniformly for families of celerities with a common bound on their total variation.
The paper leaves as an open problem the question of non-concentration in the case of a function $\tilde{c}y)$ which is continuous but not of bounded variation.
Mots clés
- eigenfunctions
- second-order elliptic
- exponential decay
- well of profile
- bounded variation
- piecewise constant
- celerity
- divergence form
- July 10 2025. 2010 Mathematics Subject Classification. Primary 35J25
- layered media
- non-concentration
- 58J50 concentration
- 2025. 2010 Mathematics Subject Classification. Primary 35J25
- July 10
- Secondary 35P20 58J50 concentration non-concentration layered media eigenfunctions second-order elliptic divergence form celerity piecewise constant bounded variation well of profile exponential decay
- Secondary 35P20
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