TY - JOUR

T1 - Homogenization for a periodic elliptic operator in a strip with various boundary conditions

AU - Senik, N. N.

PY - 2014

Y1 - 2014

N2 - A homogenization problem is considered for the periodic elliptic differential operators on $ L_2(\Pi )$, $ \Pi =\mathbb{R} \times (0, a)$, defined by the differential expression $\displaystyle \mathcal {B}_{\lambda }^{\varepsilon } = \sum _{j=1}^2 \mathrm {D... ...repsilon , x_2)\mathrm {D}_j + \mathrm {D}_j h_j(x_1/\varepsilon , x_2) \bigr )$ $\displaystyle + \ Q(x_1/\varepsilon , x_2) + \lambda Q_*(x_1/\varepsilon x_2)$ with periodic, Neumann, or Dirichlet boundary conditions. The coefficients of the expression are assumed to be periodic of period $ 1$ in the first variable and smooth in some sense in the second. Sharp-order approximations are obtained for the inverse of $ \mathcal {B}_{\lambda }^{\varepsilon }$ with respect to $ \mathbf {B}\bigl ( L_2(\Pi )\bigr )$- and $ \mathbf {B}\bigl ( L_2(\Pi ), H^1(\Pi ) \bigr )$-norms in the small $ \varepsilon $ limit with error terms of order $ \varepsilon $.

AB - A homogenization problem is considered for the periodic elliptic differential operators on $ L_2(\Pi )$, $ \Pi =\mathbb{R} \times (0, a)$, defined by the differential expression $\displaystyle \mathcal {B}_{\lambda }^{\varepsilon } = \sum _{j=1}^2 \mathrm {D... ...repsilon , x_2)\mathrm {D}_j + \mathrm {D}_j h_j(x_1/\varepsilon , x_2) \bigr )$ $\displaystyle + \ Q(x_1/\varepsilon , x_2) + \lambda Q_*(x_1/\varepsilon x_2)$ with periodic, Neumann, or Dirichlet boundary conditions. The coefficients of the expression are assumed to be periodic of period $ 1$ in the first variable and smooth in some sense in the second. Sharp-order approximations are obtained for the inverse of $ \mathcal {B}_{\lambda }^{\varepsilon }$ with respect to $ \mathbf {B}\bigl ( L_2(\Pi )\bigr )$- and $ \mathbf {B}\bigl ( L_2(\Pi ), H^1(\Pi ) \bigr )$-norms in the small $ \varepsilon $ limit with error terms of order $ \varepsilon $.

KW - homogenization

KW - operator error estimates

KW - periodic differential operators

KW - effective operator

KW - corrector

U2 - 10.1090/S1061-0022-2014-01311-8

DO - 10.1090/S1061-0022-2014-01311-8

M3 - Article

VL - 25

SP - 647

EP - 697

JO - St. Petersburg Mathematical Journal

JF - St. Petersburg Mathematical Journal

SN - 1061-0022

IS - 4

ER -