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

Keywords: singularly perturbed operators, eigenpair asymptotics, homogenization

@article{COCV_2014__20_4_1059_0, author = {Piatnitski, A. and Rybalko, A. and Rybalko, V.}, title = {Ground states of singularly perturbed convection-diffusion equation with oscillating coefficients}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {1059--1077}, publisher = {EDP-Sciences}, volume = {20}, number = {4}, year = {2014}, doi = {10.1051/cocv/2014007}, mrnumber = {3264234}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2014007/} }

TY - JOUR AU - Piatnitski, A. AU - Rybalko, A. AU - Rybalko, V. TI - Ground states of singularly perturbed convection-diffusion equation with oscillating coefficients JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2014 DA - 2014/// SP - 1059 EP - 1077 VL - 20 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2014007/ UR - https://www.ams.org/mathscinet-getitem?mr=3264234 UR - https://doi.org/10.1051/cocv/2014007 DO - 10.1051/cocv/2014007 LA - en ID - COCV_2014__20_4_1059_0 ER -

%0 Journal Article %A Piatnitski, A. %A Rybalko, A. %A Rybalko, V. %T Ground states of singularly perturbed convection-diffusion equation with oscillating coefficients %J ESAIM: Control, Optimisation and Calculus of Variations %D 2014 %P 1059-1077 %V 20 %N 4 %I EDP-Sciences %U https://doi.org/10.1051/cocv/2014007 %R 10.1051/cocv/2014007 %G en %F COCV_2014__20_4_1059_0

Piatnitski, A.; Rybalko, A.; Rybalko, V. Ground states of singularly perturbed convection-diffusion equation with oscillating coefficients. ESAIM: Control, Optimisation and Calculus of Variations, Volume 20 (2014) no. 4, pp. 1059-1077. doi : 10.1051/cocv/2014007. http://archive.numdam.org/articles/10.1051/cocv/2014007/

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