American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Weak solutions for a doubly degenerate quasilinear parabolic equation with random forcing
  • DCDS-B Home
  • This Issue
  • Next Article
    Phase transitions in a coupled viscoelastic system with periodic initial-boundary condition: (II) Convergence
June  2007, 7(4): 859-883. doi: 10.3934/dcdsb.2007.7.859

On periodic Steklov type eigenvalue problems on half-bands and the spectral homogenization problem

Eugenia Pérez 1,

1. 

Departamento de Matemática Aplicada y Ciencias de la Computación, Universided de Cantabria, Avenida de los Castros s/n, 39005 Santander, Spain

Received  May 2006 Revised  December 2006 Published  March 2007

We study the asymptotic behavior of the eigenvalues $\beta^\varepsilon$ and the associated eigenfunctions of an $\varepsilon$-dependent Steklov type eigenvalue problem posed in a bounded domain $\Omega$ of $\R^2$, when $\varepsilon \to 0$. The eigenfunctions $u^\varepsilon$ being harmonic functions inside $\Omega$, the Steklov condition is imposed on segments $T^\varepsilon$ of length $O(\varepsilon)$ periodically distributed on a fixed part $\Sigma$ of the boundary $\partial \Omega$; a homogeneous Dirichlet condition is imposed outside. The homogenization of this problem as $\varepsilon \to 0$ involves the study of the spectral local problem posed in the unit reference domain, namely the half-band $G=(-P/2,P/2)\times (0,+\infty)$ with $P$ a fixed number, with periodic conditions on the lateral boundaries and mixed boundary conditions of Dirichlet and Steklov type respectively on the segment lying on $\{y_2=0\}$. We characterize the asymptotic behavior of the low frequencies of the homogenization problem, namely of $\beta^\varepsilon\varepsilon$, and the associated eigenfunctions by means of those of the local problem.
Keywords: boundary homogenization, Spectral analysis, Steklov eigenvalue problems, low frequencies..
Mathematics Subject Classification: Primary: 35B27, 35P05, 47F05, 35P15; Secondary: 86A15, 47A55, 47B3.
Citation: Eugenia Pérez. On periodic Steklov type eigenvalue problems on half-bands and the spectral homogenization problem. Discrete & Continuous Dynamical Systems - B, 2007, 7 (4) : 859-883. doi: 10.3934/dcdsb.2007.7.859
[1]

Toshiyuki Ogawa, Takashi Okuda. Bifurcation analysis to Swift-Hohenberg equation with Steklov type boundary conditions. Discrete & Continuous Dynamical Systems, 2009, 25 (1) : 273-297. doi: 10.3934/dcds.2009.25.273
[2]

Elvise Berchio, Filippo Gazzola, Dario Pierotti. Nodal solutions to critical growth elliptic problems under Steklov boundary conditions. Communications on Pure & Applied Analysis, 2009, 8 (2) : 533-557. doi: 10.3934/cpaa.2009.8.533
[3]

Wanbin Tong, Hongjin He, Chen Ling, Liqun Qi. A nonmonotone spectral projected gradient method for tensor eigenvalue complementarity problems. Numerical Algebra, Control & Optimization, 2020, 10 (4) : 425-437. doi: 10.3934/naco.2020042
[4]

Natalia O. Babych, Ilia V. Kamotski, Valery P. Smyshlyaev. Homogenization of spectral problems in bounded domains with doubly high contrasts. Networks & Heterogeneous Media, 2008, 3 (3) : 413-436. doi: 10.3934/nhm.2008.3.413
[5]

J. R. L. Webb. Uniqueness of the principal eigenvalue in nonlocal boundary value problems. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 177-186. doi: 10.3934/dcdss.2008.1.177
[6]

Bruno Colbois, Alexandre Girouard. The spectral gap of graphs and Steklov eigenvalues on surfaces. Electronic Research Announcements, 2014, 21: 19-27. doi: 10.3934/era.2014.21.19
[7]

Germain Gendron. Uniqueness results in the inverse spectral Steklov problem. Inverse Problems & Imaging, 2020, 14 (4) : 631-664. doi: 10.3934/ipi.2020029
[8]

Sergei Avdonin, Fritz Gesztesy, Konstantin A. Makarov. Spectral estimation and inverse initial boundary value problems. Inverse Problems & Imaging, 2010, 4 (1) : 1-9. doi: 10.3934/ipi.2010.4.1
[9]

Rafael Abreu, Cristian Morales-Rodrigo, Antonio Suárez. Some eigenvalue problems with non-local boundary conditions and applications. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2465-2474. doi: 10.3934/cpaa.2014.13.2465
[10]

Gloria Paoli, Gianpaolo Piscitelli, Rossanno Sannipoli. A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle. Communications on Pure & Applied Analysis, 2021, 20 (1) : 145-158. doi: 10.3934/cpaa.2020261
[11]

Monika Laskawy. Optimality conditions of the first eigenvalue of a fourth order Steklov problem. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1843-1859. doi: 10.3934/cpaa.2017089
[12]

Ya Li, ShouQiang Du, YuanYuan Chen. Modified spectral PRP conjugate gradient method for solving tensor eigenvalue complementarity problems. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020147
[13]

Grégoire Allaire, Yves Capdeboscq, Marjolaine Puel. Homogenization of a one-dimensional spectral problem for a singularly perturbed elliptic operator with Neumann boundary conditions. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 1-31. doi: 10.3934/dcdsb.2012.17.1
[14]

Farah Abdallah, Denis Mercier, Serge Nicaise. Spectral analysis and exponential or polynomial stability of some indefinite sign damped problems. Evolution Equations & Control Theory, 2013, 2 (1) : 1-33. doi: 10.3934/eect.2013.2.1
[15]

Valeria Chiado Piat, Sergey S. Nazarov, Andrey Piatnitski. Steklov problems in perforated domains with a coefficient of indefinite sign. Networks & Heterogeneous Media, 2012, 7 (1) : 151-178. doi: 10.3934/nhm.2012.7.151
[16]

Carlos J. S. Alves, Nuno F. M. Martins, Nilson C. Roberty. Full identification of acoustic sources with multiple frequencies and boundary measurements. Inverse Problems & Imaging, 2009, 3 (2) : 275-294. doi: 10.3934/ipi.2009.3.275
[17]

Nikolaos S. Papageorgiou, Vicenţiu D. Rădulescu, Dušan D. Repovš. Perturbations of nonlinear eigenvalue problems. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1403-1431. doi: 10.3934/cpaa.2019068
[18]

Maria Fărcăşeanu, Mihai Mihăilescu, Denisa Stancu-Dumitru. Perturbed fractional eigenvalue problems. Discrete & Continuous Dynamical Systems, 2017, 37 (12) : 6243-6255. doi: 10.3934/dcds.2017270
[19]

Ravi P. Agarwal, Kanishka Perera, Zhitao Zhang. On some nonlocal eigenvalue problems. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 707-714. doi: 10.3934/dcdss.2012.5.707
[20]

Lujuan Yu. The asymptotic behaviour of the $ p(x) $-Laplacian Steklov eigenvalue problem. Discrete & Continuous Dynamical Systems - B, 2020, 25 (7) : 2621-2637. doi: 10.3934/dcdsb.2020025

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (63)
  • HTML views (0)
  • Cited by (14)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences