2010, 9(2): 539-561. doi: 10.3934/cpaa.2010.9.539

Asymptotic profiles of eigenfunctions for some 1-dimensional linearized eigenvalue problems

1. 

Department of Applied Mathematics, Waseda University, Ohkubo, Shinjuku-ku, Tokyo 169-8555, Japan

2. 

Department of Applied Mathematics and Informatics, Ryukoku University, Seta, Otsu, Shiga 520-2194, Japan

Received  January 2009 Revised  October 2009 Published  December 2009

We are interested in the asymptotic profiles of all eigenfunctions for 1-dimensional linearized eigenvalue problems to nonlinear boundary value problems with a diffusion coefficient $\varepsilon$. For instance, it seems that they have simple and beautiful properties for sufficiently small $\varepsilon$ in the balanced bistable nonlinearity case. As the first step to give rigorous proofs for the above case, we study the case $f(u)=\sin u$ precisely. We show that two special eigenfunctions completely control the asymptotic profiles of other eigenfunctions.
Citation: Tohru Wakasa, Shoji Yotsutani. Asymptotic profiles of eigenfunctions for some 1-dimensional linearized eigenvalue problems. Communications on Pure & Applied Analysis, 2010, 9 (2) : 539-561. doi: 10.3934/cpaa.2010.9.539
[1]

Giuseppina Barletta, Roberto Livrea, Nikolaos S. Papageorgiou. A nonlinear eigenvalue problem for the periodic scalar $p$-Laplacian. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1075-1086. doi: 10.3934/cpaa.2014.13.1075

[2]

Isabeau Birindelli, Stefania Patrizi. A Neumann eigenvalue problem for fully nonlinear operators. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 845-863. doi: 10.3934/dcds.2010.28.845

[3]

Tohru Wakasa, Shoji Yotsutani. Representation formulas for some 1-dimensional linearized eigenvalue problems. Communications on Pure & Applied Analysis, 2008, 7 (4) : 745-763. doi: 10.3934/cpaa.2008.7.745

[4]

Li-Fang Dai, Mao-Lin Liang, Wei-Yuan Ma. Optimization problems on the rank of the solution to left and right inverse eigenvalue problem. Journal of Industrial & Management Optimization, 2015, 11 (1) : 171-183. doi: 10.3934/jimo.2015.11.171

[5]

Jonathan E. Rubin. A nonlocal eigenvalue problem for the stability of a traveling wave in a neuronal medium. Discrete & Continuous Dynamical Systems - A, 2004, 10 (4) : 925-940. doi: 10.3934/dcds.2004.10.925

[6]

Mihai Mihăilescu. An eigenvalue problem possessing a continuous family of eigenvalues plus an isolated eigenvalue. Communications on Pure & Applied Analysis, 2011, 10 (2) : 701-708. doi: 10.3934/cpaa.2011.10.701

[7]

Giacomo Bocerani, Dimitri Mugnai. A fractional eigenvalue problem in $\mathbb{R}^N$. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 619-629. doi: 10.3934/dcdss.2016016

[8]

David Colton, Yuk-J. Leung. On a transmission eigenvalue problem for a spherically stratified coated dielectric. Inverse Problems & Imaging, 2016, 10 (2) : 369-378. doi: 10.3934/ipi.2016004

[9]

Jean-Michel Rakotoson. Generalized eigenvalue problem for totally discontinuous operators. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 343-373. doi: 10.3934/dcds.2010.28.343

[10]

Huan Gao, Zhibao Li, Haibin Zhang. A fast continuous method for the extreme eigenvalue problem. Journal of Industrial & Management Optimization, 2017, 13 (3) : 1587-1599. doi: 10.3934/jimo.2017008

[11]

Yinnian He, Yi Li. Asymptotic behavior of linearized viscoelastic flow problem. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 843-856. doi: 10.3934/dcdsb.2008.10.843

[12]

Fioralba Cakoni, Houssem Haddar, Isaac Harris. Homogenization of the transmission eigenvalue problem for periodic media and application to the inverse problem. Inverse Problems & Imaging, 2015, 9 (4) : 1025-1049. doi: 10.3934/ipi.2015.9.1025

[13]

Teodora-Liliana Dinu. Entire solutions of the nonlinear eigenvalue logistic problem with sign-changing potential and absorption. Communications on Pure & Applied Analysis, 2003, 2 (3) : 311-321. doi: 10.3934/cpaa.2003.2.311

[14]

VicenŢiu D. RǍdulescu, Somayeh Saiedinezhad. A nonlinear eigenvalue problem with $ p(x) $-growth and generalized Robin boundary value condition. Communications on Pure & Applied Analysis, 2018, 17 (1) : 39-52. doi: 10.3934/cpaa.2018003

[15]

Xing-Bin Pan. An eigenvalue variation problem of magnetic Schrödinger operator in three dimensions. Discrete & Continuous Dynamical Systems - A, 2009, 24 (3) : 933-978. doi: 10.3934/dcds.2009.24.933

[16]

Chiu-Yen Kao, Yuan Lou, Eiji Yanagida. Principal eigenvalue for an elliptic problem with indefinite weight on cylindrical domains. Mathematical Biosciences & Engineering, 2008, 5 (2) : 315-335. doi: 10.3934/mbe.2008.5.315

[17]

Aixia Qian, Shujie Li. Multiple sign-changing solutions of an elliptic eigenvalue problem. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 737-746. doi: 10.3934/dcds.2005.12.737

[18]

Yafeng Li, Guo Sun, Yiju Wang. A smoothing Broyden-like method for polyhedral cone constrained eigenvalue problem. Numerical Algebra, Control & Optimization, 2011, 1 (3) : 529-537. doi: 10.3934/naco.2011.1.529

[19]

Bruce Geist and Joyce R. McLaughlin. Eigenvalue formulas for the uniform Timoshenko beam: the free-free problem. Electronic Research Announcements, 1998, 4: 12-17.

[20]

Huan Gao, Yu-Hong Dai, Xiao-Jiao Tong. Barzilai-Borwein-like methods for the extreme eigenvalue problem. Journal of Industrial & Management Optimization, 2015, 11 (3) : 999-1019. doi: 10.3934/jimo.2015.11.999

2016 Impact Factor: 0.801

Metrics

  • PDF downloads (0)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]