TY - JOUR

T1 - HOMOGENIZATION FOR NON-SELF-ADJOINT PERIODIC ELLIPTIC OPERATORS ON AN INFINITE CYLINDER

AU - Senik, N.N.

PY - 2017

Y1 - 2017

N2 - © 2017 Nikita N. Senik.We consider the problem of homogenization for non-self-adjoint second-order elliptic differential operators Aε of divergence form on L2 (ℝd1 × double-struck Td2), where d1 is positive and d2 is non-negative. The coefficients of the operator Aε are periodic in the first variable with period ε and smooth in a certain sense in the second. We show that, as ε gets small, (Aε -μ)-1 and ∇x2 (Aε -μ)-1 for an appropriate μ converge in the operator norm to, respectively, (A0 - μ)-1 and ∇x2 (A0 - μ)-1, where A0 is an operator whose coefficients depend only on x2. We also obtain an approximation for ∇x1 (Aε - μ)-1 and find the next term in the approximation for (Aε - μ)-1 . Estimates for the rates of convergence and the rates of approximation are provided and are sharp with respect to the order.

AB - © 2017 Nikita N. Senik.We consider the problem of homogenization for non-self-adjoint second-order elliptic differential operators Aε of divergence form on L2 (ℝd1 × double-struck Td2), where d1 is positive and d2 is non-negative. The coefficients of the operator Aε are periodic in the first variable with period ε and smooth in a certain sense in the second. We show that, as ε gets small, (Aε -μ)-1 and ∇x2 (Aε -μ)-1 for an appropriate μ converge in the operator norm to, respectively, (A0 - μ)-1 and ∇x2 (A0 - μ)-1, where A0 is an operator whose coefficients depend only on x2. We also obtain an approximation for ∇x1 (Aε - μ)-1 and find the next term in the approximation for (Aε - μ)-1 . Estimates for the rates of convergence and the rates of approximation are provided and are sharp with respect to the order.

KW - homogenization

KW - operator error estimates

KW - periodic differential operators

KW - effective operator

KW - corrector

UR - https://epubs.siam.org/doi/10.1137/15M1049981

UR - https://doi.org/10.1137/15M1049981

U2 - 10.1137/15M1049981

DO - 10.1137/15M1049981

M3 - Article

VL - 49

SP - 874

EP - 898

JO - SIAM Journal on Mathematical Analysis

JF - SIAM Journal on Mathematical Analysis

SN - 0036-1410

IS - 2

ER -