Concentration and non-concentration of eigenfunctions of second-order elliptic operators with a divergence form in layered media

Abstract

Let $\Omega$ ' $\subset$ R^d , d = 1, 2, . . . be an open bounded smooth domain, and $\Omega = \Omega'\times (0,H)\subset \mathbb{R}^d \times \mathbb{R}_+.$ The coordinates in $\Omega$ are designated as x = (x ' , y) $\in$ $\Omega$ ' x (0, H). The paper deals with the concentration (and non-concentration) properties (in sectors of $\Omega$) 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 $\in$ (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.