TY - JOUR

T1 - On Homogenization for Non-Self-Adjoint Periodic Elliptic Operators on an Infinite Cylinder

AU - Senik, N. N.

PY - 2016

Y1 - 2016

N2 - We consider an operator~$\mathcal{A}^{\varepsilon}$ on $L_{2}(\mathbb{R}^{d_{1}}\times\mathbb{T}^{d_{2}})$ ($d_{1}$~is positive, while $d_{2}$ can be zero) given by $\mathcal{A}^{\varepsilon}=D^{*}A\rbr{\varepsilon^{-1}x_{1},x_{2}}\,D$ where $A$ is periodic in the first variable and smooth in a sense in the second. We present approximations for $(\mathcal{A}^{\varepsilon}-\mu)^{-1}$ and~$D(\mathcal{A}^{\varepsilon}-\mu)^{-1}$ (with an appropriate~$\mu$) in the operator norm when $\varepsilon$ is small. We also provide estimates for the rates of approximation that are sharp with respect to the order.

AB - We consider an operator~$\mathcal{A}^{\varepsilon}$ on $L_{2}(\mathbb{R}^{d_{1}}\times\mathbb{T}^{d_{2}})$ ($d_{1}$~is positive, while $d_{2}$ can be zero) given by $\mathcal{A}^{\varepsilon}=D^{*}A\rbr{\varepsilon^{-1}x_{1},x_{2}}\,D$ where $A$ is periodic in the first variable and smooth in a sense in the second. We present approximations for $(\mathcal{A}^{\varepsilon}-\mu)^{-1}$ and~$D(\mathcal{A}^{\varepsilon}-\mu)^{-1}$ (with an appropriate~$\mu$) in the operator norm when $\varepsilon$ is small. We also provide estimates for the rates of approximation that are sharp with respect to the order.

KW - homogenization

KW - operator error estimates

KW - periodic differential operators

KW - effective operator

KW - corrector

U2 - 10.1007/s10688-016-0131-6

DO - 10.1007/s10688-016-0131-6

M3 - Article

VL - 50

SP - 71

EP - 75

JO - Functional Analysis and its Applications

JF - Functional Analysis and its Applications

SN - 0016-2663

IS - 1

ER -