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Wave propagation in guiding structures / Propagation d'ondes dans les structures guidées

Collection Wave propagation in guiding structures / Propagation d'ondes dans les structures guidées

Organizer(s) Briet, Philippe ; Cassier, Maxence ; Cossetti, Lucrezia ; Ourmières-Bonafos, Thomas ; Verri, Alessandra
Date(s) 27/10/2025 - 31/10/2025
linked URL https://conferences.cirm-math.fr/3333.html
00:00:00 / 00:00:00
5 7

Dimension reduction in thin domains - Lecture 2

By Alessandra Verri

This lecture aims to introduce students to the idea of dimension reduction, highlighting basic strategies presented in the literature. In the first part, we address challenges and fundamental approaches in the study of the Dirichlet Laplacian on thin two-dimensional strips. Three models are considered—straight strips, curved strips, and strips with non-uniform width—illustrating how geometry influences the asymptotic behavior of eigenvalues as the width tends to zero. The corresponding effective operators obtained in this limit include the one-dimensional Laplacian and Schrödinger-type operators with geometric potentials.In the second part, we extend the discussion to strips embedded in three-dimensional space and equipped with mixed Dirichlet–Neumann boundary conditions. These results are inspired by and build upon previous works on the purely Dirichlet case, which serve as the main reference for our analysis. A comparison with the Dirichlet setting highlights both the analogies and the differences in the asymptotic behavior of eigenvalues as the strip becomes thin.

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Citation data

  • DOI 10.24350/CIRM.V.20402503
  • Cite this video Verri, Alessandra (27/10/2025). Dimension reduction in thin domains - Lecture 2. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.20402503
  • URL https://dx.doi.org/10.24350/CIRM.V.20402503

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Bibliography

  • AMORIM, Rafael T. et VERRI, Alessandra A. Spectral analysis on ruled surfaces with combined Dirichlet and Neumann boundary conditions. Journal of Mathematical Physics, 2023, vol. 64, no 10. - https://doi.org/10.1063/5.0099904
  • DUCLOS, Pierre et EXNER, Pavel. Curvature-induced bound states in quantum waveguides in two and three dimensions. Reviews in Mathematical Physics, 1995, vol. 7, no 01, p. 73-102. - https://doi.org/10.1142/S0129055X95000062
  • FRIEDLANDER, Leonid et SOLOMYAK, Michael. On the spectrum of the Dirichlet Laplacian in a narrow strip. Israel journal of mathematics, 2009, vol. 170, no 1, p. 337-354. - https://doi.org/10.1007/s11856-009-0032-y
  • KREJČIŘÍK, David. Spectrum of the Laplacian in a narrow curved strip with combined Dirichlet and Neumann boundary conditions. ESAIM: Control, Optimisation and Calculus of Variations, 2009, vol. 15, no 3, p. 555-568. - https://doi.org/10.1051/cocv:2008035
  • KREJČIŘÍK, D. et ZAHRADOVÁ, K. Quantum strips in higher dimensions. Oper. Matrices, 2020, vol. 14, no 3, p. 635-665. - https://doi.org/10.7153/oam-2020-14-41

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