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Ground states of singularly perturbed convection-diffusionequation with oscillating coefficients

Published online by Cambridge University Press:  04 August 2014

A. Piatnitski ,
A. Rybalko  and
V. Rybalko
A. Piatnitski
Affiliation:
Narvik University College, Postboks 385, 8505 Narvik, Norway, and P.N. Lebedev Physical Institute of RAS, 53, Leninski pr., 119991 Moscow, Russia. andrey@sci.lebedev.ru
A. Rybalko
Affiliation:
Kharkiv National University of Economics, 9a Lenin ave., 61166 Kharkiv, Ukraine; nrybalko@yahoo.com
V. Rybalko
Affiliation:
Mathematical Department, B.Verkin Institute for Low Temperature Physics and Engineering of the NASU, 47 Lenin ave., 61103 Kharkiv, Ukraine; vrybalko@ilt.kharkov.ua ,
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Abstract

We study the first eigenpair of a Dirichlet spectral problem for singularly perturbedconvection-diffusion operators with oscillating locally periodic coefficients. It followsfrom the results of [A. Piatnitski and V. Rybalko, On the first eigenpair of singularlyperturbed operators with oscillating coefficients. Preprintwww.arxiv.org, arXiv:1206.3754] that thefirst eigenvalue remains bounded only if the integral curves of the so-called effectivedrift have a nonempty ω-limit set. Here we consider the case when theintegral curves can have both hyperbolic fixed points and hyperbolic limit cycles. One ofthe main goals of this work is to determine a fixed point or a limit cycle responsible forthe first eigenpair asymptotics. Here we focus on the case of limit cycles that was leftopen in [A. Piatnitski and V. Rybalko, Preprint.

Keywords

Singularly perturbed operatorseigenpair asymptoticshomogenization
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 20 , Issue 4 , October 2014 , pp. 1059 - 1077
Copyright
© EDP Sciences, SMAI, 2014

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