Geodesic
Singularly perturbed spectral problems in a thin cylinder with Fourier conditions on its bases
Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 15 (2019) no. 2, pp. 256-277 
Andrey Piatnitski; Volodymyr Rybalko. Singularly perturbed spectral problems in a thin cylinder with Fourier conditions on its bases. Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 15 (2019) no. 2, pp. 256-277. http://geodesic.mathdoc.fr/item/JMAG_2019_15_2_a6/
@article{JMAG_2019_15_2_a6,
     author = {Andrey Piatnitski and Volodymyr Rybalko},
     title = {Singularly perturbed spectral problems in a thin cylinder with {Fourier} conditions on its bases},
     journal = {\v{Z}urnal matemati\v{c}eskoj fiziki, analiza, geometrii},
     pages = {256--277},
     year = {2019},
     volume = {15},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JMAG_2019_15_2_a6/}
}
TY  - JOUR
AU  - Andrey Piatnitski
AU  - Volodymyr Rybalko
TI  - Singularly perturbed spectral problems in a thin cylinder with Fourier conditions on its bases
JO  - Žurnal matematičeskoj fiziki, analiza, geometrii
PY  - 2019
SP  - 256
EP  - 277
VL  - 15
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/JMAG_2019_15_2_a6/
LA  - en
ID  - JMAG_2019_15_2_a6
ER  - 
%0 Journal Article
%A Andrey Piatnitski
%A Volodymyr Rybalko
%T Singularly perturbed spectral problems in a thin cylinder with Fourier conditions on its bases
%J Žurnal matematičeskoj fiziki, analiza, geometrii
%D 2019
%P 256-277
%V 15
%N 2
%U http://geodesic.mathdoc.fr/item/JMAG_2019_15_2_a6/
%G en
%F JMAG_2019_15_2_a6

Voir la notice de l'article provenant de la source Math-Net.Ru

The paper deals with the bottom of the spectrum of a singularly perturbed second order elliptic operator defined in a thin cylinder and having locally periodic coefficients in the longitudinal direction. We impose a homogeneous Neumann boundary condition on the lateral surface of the cylinder and a generic homogeneous Fourier condition at its bases. We then show that the asymptotic behavior of the principal eigenpair can be characterized in terms of the limit one-dimensional problem for the effective Hamilton–Jacobi equation with the effective boundary conditions. In order to construct boundary layer correctors we study a Steklov type spectral problem in a semi-infinite cylinder (these results are of independent interest). Under a structure assumption on the effective problem leading to localization (in certain sense) of eigenfunctions inside the cylinder we prove a two-term asymptotic formula for the first and higher order eigenvalues.

[1] G. Allaire, Y. Capdeboscq, M. Puel, “Homogenization of a one-dimensional spectral problem for a singularly perturbed elliptic operator with Neumann boundary conditions”, Discrete Contin. Dyn. Syst. Ser. B, 17:1 (2012), 1–31 | MR | Zbl

[2] G. Allaire, A. Piatnitski, “On the asymptotic behaviour of the kernel of an adjoint convection-diffusion operator in a long cylinder”, Rev. Mat. Iberoam., 33:4 (2017), 1123–1148 | DOI | MR | Zbl

[3] M. Arisawa, “Long time averaged reflection force and homogenization of oscillating Neumann boundary conditions”, Ann. Inst. H. Poincaré (C) Anal. Non Linéaire, 20:2 (2003), 293–332 | DOI | MR | Zbl

[4] G. Barles, F. Da Lio, P.-L. Lions, P.E. Souganidis, “Ergodic problems and periodic homogenization for fully nonlinear equations in half-space type domains with Neumann boundary conditions”, Indiana Univ. Math. J., 57:5 (2008), 2355–2375 | DOI | MR | Zbl

[5] I. Capuzzo-Dolcetta, P.-L. Lions, “Hamilton–Jacobi equations with state constraints”, Trans. Amer. Math. Soc., 318 (1990), 643–683 | DOI | MR | Zbl

[6] D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin–Heidelberg–New York–Tokyo, 1983 | MR | Zbl

[7] M.G. Crandall, H. Ishii, P.-L. Lions, “User's guide to viscosity solutions of second order partial dierential equations”, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1–67 | DOI | MR | Zbl

[8] L.C. Evans, “The perturbed test function method for viscosity solutions of nonlinear PDE”, Proc. Roy. Soc. Edinburgh Sect. A, 111:3–4 (1989), 359–375 | DOI | MR | Zbl

[9] G.M. Lieberman, “Oblique derivative problems in Lipschitz domains. II. Discontinuous boundary data”, J. Reine Angew. Math., 389 (1988), 1–21 | MR | Zbl

[10] I. Pankratova, A. Piatnitski, “On behavior at infinity of solutions to stationary convection-diffusion equation in a cylinder”, Discrete Contin. Dyn. Syst. Ser. B, 11:4 (2009), 935–970 | MR | Zbl

[11] A. Piatnitski, V. Rybalko, “On the first eigenpair of singularly perturbed operators with oscillating coefficients”, Comm. Partial Differential Equations, 41:1 (2016), 1–31 | DOI | MR | Zbl

[12] A. Piatnitski, A. Rybalko, V. Rybalko, “Ground states of singularly perturbed convection-diffusion equation with oscillating coefficients”, ESAIM Control Optim. Calc. Var., 20:4 (2014), 1059–1077 | DOI | MR | Zbl

[13] A. Piatnitski, A. Rybalko, V. Rybalko, “Singularly perturbed spectral problems with Neumann boundary conditions”, Complex Var. Elliptic Equ., 61:2 (2015), 252–274 | DOI | MR

[14] H.M. Soner, “Optimal control with state-space constraint. I”, SIAM J. Control Optim., 24 (1986), 552–561 | DOI | MR | Zbl