Concentration and non-concentration of eigenfunctions of second-order elliptic operators with a divergence form in layered media
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2025
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| author | Ben-Artzi, Matania Dermenjian, Yves |
| author_facet | Ben-Artzi, Matania Dermenjian, Yves |
| contents | Let $Ω$ ' $\subset$ R^d , d = 1, 2, . . . be an open bounded smooth domain, and $Ω= Ω'\times (0,H)\subset \mathbb{R}^d \times \mathbb{R}_+.$ The coordinates in $Ω$ are designated as x = (x ' , y) $\in$ $Ω$ ' x (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 $L^2(Ω,dx)$ with domain $D(A) = \{v\in H_0^1(Ω); \tilde{c}\nabla v \in H^1(Ω)\}.$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_06352 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Concentration and non-concentration of eigenfunctions of second-order elliptic operators with a divergence form in layered media Ben-Artzi, Matania Dermenjian, Yves Algebraic Geometry Let $Ω$ ' $\subset$ R^d , d = 1, 2, . . . be an open bounded smooth domain, and $Ω= Ω'\times (0,H)\subset \mathbb{R}^d \times \mathbb{R}_+.$ The coordinates in $Ω$ are designated as x = (x ' , y) $\in$ $Ω$ ' x (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 $L^2(Ω,dx)$ with domain $D(A) = \{v\in H_0^1(Ω); \tilde{c}\nabla v \in H^1(Ω)\}.$ 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. |
| title | Concentration and non-concentration of eigenfunctions of second-order elliptic operators with a divergence form in layered media |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2509.06352 |