TY - JOUR

T1 - On Homogenization of Locally Periodic Elliptic and Parabolic Operators

AU - Senik, N. N.

N1 - Publisher Copyright: © 2020, Pleiades Publishing, Ltd.

PY - 2020/1/1

Y1 - 2020/1/1

N2 - Let Ω be a C1,s bounded domain (s > 1/2) in ℝd, and let Aε= − div A(x, x/ ε) ∇ be a matrix elliptic operator on Ω with Dirichlet boundary condition. We suppose that ε is small and the function A is Lipschitz in the first variable and periodic in the second one, so the coefficients of Aε are locally periodic. For μ in the resolvent set, we are interested in finding the rates of approximations, as ε → 0, for (Aε−μρε)−1 and ∇(Aε−μρε)−1 in the operator topology on L2. Here ρε(x)= ρ(x,x/ε) is a positive definite locally periodic function with ρ satisfying the same assumptions as A. Keeping track of the rate dependence on both ε and μ, we then proceed to similar questions for the solution to the initial boundary-value problem ρε∂ tvε= − Aεvε.

AB - Let Ω be a C1,s bounded domain (s > 1/2) in ℝd, and let Aε= − div A(x, x/ ε) ∇ be a matrix elliptic operator on Ω with Dirichlet boundary condition. We suppose that ε is small and the function A is Lipschitz in the first variable and periodic in the second one, so the coefficients of Aε are locally periodic. For μ in the resolvent set, we are interested in finding the rates of approximations, as ε → 0, for (Aε−μρε)−1 and ∇(Aε−μρε)−1 in the operator topology on L2. Here ρε(x)= ρ(x,x/ε) is a positive definite locally periodic function with ρ satisfying the same assumptions as A. Keeping track of the rate dependence on both ε and μ, we then proceed to similar questions for the solution to the initial boundary-value problem ρε∂ tvε= − Aεvε.

KW - elliptic systems

KW - homogenization

KW - locally periodic operators

KW - operator error estimates

KW - parabolic systems

UR - http://www.scopus.com/inward/record.url?scp=85090090052&partnerID=8YFLogxK

U2 - 10.1134/S0016266320010104

DO - 10.1134/S0016266320010104

M3 - Article

AN - SCOPUS:85090090052

VL - 54

SP - 68

EP - 72

JO - Functional Analysis and its Applications

JF - Functional Analysis and its Applications

SN - 0016-2663

IS - 1

ER -