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

Tohru Wakasa; Shoji Yotsutani

doi:10.3934/cpaa.2010.9.539

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2010, Volume 9, Issue 2: 539-561. Doi: 10.3934/cpaa.2010.9.539

This issue Previous Article Stokes-Brinkman transmission problems on Lipschitz and $C^1$ domains in Riemannian manifolds Next Article Semi-positone nonlocal boundary value problems of arbitrary order

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

Tohru Wakasa^1, and
Shoji Yotsutani^2,

1.
Department of Applied Mathematics, Waseda University, Ohkubo, Shinjuku-ku, Tokyo 169-8555
2.
Department of Applied Mathematics and Informatics, Ryukoku University, Seta, Otsu, Shiga 520-2194

Received: December 31, 2008

Revised: September 30, 2009

Published: February 2010

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Abstract

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.

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Mathematics Subject Classification: 34A05, 34B05, 34B15, 35K57.

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