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Revista Matemática Complutense

, Volume 31, Issue 1, pp 1–62 | Cite as

On multi-scale asymptotic structure of eigenfunctions in a boundary value problem with concentrated masses near the boundary

  • Sergei A. Nazarov
  • M. Eugenia PérezEmail author
Article

Abstract

We construct two-term asymptotics \(\lambda ^\varepsilon _k= \varepsilon ^{m-2}(M+\varepsilon \mu _k+ O(\varepsilon ^{3/2}) )\) of eigenvalues of a mixed boundary-value problem in \(\Omega \subset {{\mathbb {R}}}^2\) with many heavy (\(m>2\)) concentrated masses near a straight part \(\Gamma \) of the boundary \(\partial \Omega \). \(\varepsilon \) is a small positive parameter related to size and periodicity of the masses; \(k\in {\mathbb N}\). The main term \(M>0\) is common for all eigenvalues but the correction terms \(\mu _k\), which are eigenvalues of a limit problem with the spectral Steklov boundary conditions on \(\Gamma \), exhibit the effect of asymptotic splitting in the eigenvalue sequence enabling the detection of asymptotic forms of eigenfunctions. The justification scheme implies isolating and purifying singularities of eigenfunctions and leads to a new spectral problem in weighed spaces with a “strongly” singular weight.

Keywords

Spectral analysis Homogenization problems Concentrated masses Asymptotic splitting of eigenvalues Steklov problem Corner singularities 

Mathematics Subject Classification

35B25 35B27 35P05 35B40 47A75 74H45 

Notes

Acknowledgements

This research work has been partially supported by Spanish MINECO, MTM2013-44883-P. Also, the research work of the first author has been partially supported by Russian Foundation of Basic research (Project 15–01–02175).

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Copyright information

© Universidad Complutense de Madrid 2017

Authors and Affiliations

  1. 1.Department of Elasticity, Mathematics and Mechanics FacultySaint-Petersburg State UniversitySt. PetersburgRussia
  2. 2.Lababoratory of Mechanics of New Nano-MaterialsPeter the Great State Polytechnical UniversitySt. PetersburgRussia
  3. 3.Laboratory of Mathematical Methods in Mechanics of Materials, Institute of Problems of Mechanical EngineeringRASSt. PetersburgRussia
  4. 4.Departamento de Matemática Aplicada y Ciencias de la ComputaciónUniversidad de CantabriaSantanderSpain

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