On the spectrum of the Dirichlet Laplacian in a narrow strip
By L. Friedlander, Michael Solomyak. 2007
135 citations recorded by the source corpus.
We consider the Dirichlet Laplacian Δ∈ in a family of bounded domains {−a < x < b, 0 < y < εh(x)}. The main assumption is that x = 0 is the only point of global maximum of the positive, continuous function h(x). We find the two-term asymptotics in ε → 0 of the eigenvalues and the one-term asymptotics of the corresponding eigenfunctions. The asymptotic formulas obtained involve the eigenvalues and eigenfunctions of an auxiliary ODE on ℝ that depends on the behavior of h(x) as x → 0.The proof is based on a detailed study of the resolvent of the operator Δ∈.
Subfields: Computational Theory and Mathematics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/a3b92b2e8ae9e4778b1d92d8ee2c299902449307
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In Computational Theory and Mathematics
In Mathematical Physics
Citations in the corpus
- Asymptotics of Dirichlet eigenvalues and eigenfunctions of the Laplacian on thin domains in R^d (2009, methodology): “he proofs of Lemma 2.1 and Theorem 1.2 go through with minor changes (note that m= 2k, while Iand I ε should be changed by ωand ω ε, respectively). Similar remarks apply to the proofs in Section 4 of [FS] leading to the proof of Theorem 1.3, except that due to regularity we do not need to worry abou”
- Geometrical Structure of Laplacian Eigenfunctions (2012, background): “Friedlander and Solomyak studied the spectrum of the Dirichlet Laplacian in a family of narrow strips of variable profile: Ω = {(x, y) ∈ R(2) : −a < x < b, 0 < y < εh(x)} [195, 196].”