Homogenization of the first initial boundary-value problem for parabolic systems: Operator error estimates

Yu M. Meshkova, T. A. Suslina

Research output

1 Citation (Scopus)

Abstract

Let O ⊃ ℝd be a bounded domain of class C1,1. In L2(O;n), a selfadjoint matrix second order elliptic differential operator BD,ε, 0 < ε ≤ 1, is considered with the Dirichlet boundary condition. The principal part of the operator is given in a factorized form. The operator involves first and zero order terms. The operator BD,ε is positive definite; its coefficients are periodic and depend on x/ε. The behavior of the operator exponential e -BDt, t > 0, is studied as ε → 0. Approximations for the exponential e -BDt are obtained in the operator norm on L2(O;n) and in the norm of operators acting from L2(O;n) to the Sobolev space H1(O;n). The results are applied to homogenization of solutions of the first initial boundary-value problem for parabolic systems.

Original languageEnglish
Pages (from-to)935-978
Number of pages44
JournalSt. Petersburg Mathematical Journal
Volume29
Issue number6
DOIs
Publication statusPublished - 1 Jan 2018

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Sobolev spaces
Operator Norm
Parabolic Systems
Elliptic Operator
Homogenization
Initial-boundary-value Problem
Sobolev Spaces
Boundary value problems
Mathematical operators
Differential operator
Error Estimates
Bounded Domain
Norm
Approximation
Operator
Class

Scopus subject areas

  • Analysis
  • Algebra and Number Theory
  • Applied Mathematics

Cite this

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abstract = "Let O ⊃ ℝd be a bounded domain of class C1,1. In L2(O;n), a selfadjoint matrix second order elliptic differential operator BD,ε, 0 < ε ≤ 1, is considered with the Dirichlet boundary condition. The principal part of the operator is given in a factorized form. The operator involves first and zero order terms. The operator BD,ε is positive definite; its coefficients are periodic and depend on x/ε. The behavior of the operator exponential e -BD,εt, t > 0, is studied as ε → 0. Approximations for the exponential e -BD,εt are obtained in the operator norm on L2(O;n) and in the norm of operators acting from L2(O;n) to the Sobolev space H1(O;n). The results are applied to homogenization of solutions of the first initial boundary-value problem for parabolic systems.",
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T1 - Homogenization of the first initial boundary-value problem for parabolic systems

T2 - Operator error estimates

AU - Meshkova, Yu M.

AU - Suslina, T. A.

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N2 - Let O ⊃ ℝd be a bounded domain of class C1,1. In L2(O;n), a selfadjoint matrix second order elliptic differential operator BD,ε, 0 < ε ≤ 1, is considered with the Dirichlet boundary condition. The principal part of the operator is given in a factorized form. The operator involves first and zero order terms. The operator BD,ε is positive definite; its coefficients are periodic and depend on x/ε. The behavior of the operator exponential e -BD,εt, t > 0, is studied as ε → 0. Approximations for the exponential e -BD,εt are obtained in the operator norm on L2(O;n) and in the norm of operators acting from L2(O;n) to the Sobolev space H1(O;n). The results are applied to homogenization of solutions of the first initial boundary-value problem for parabolic systems.

AB - Let O ⊃ ℝd be a bounded domain of class C1,1. In L2(O;n), a selfadjoint matrix second order elliptic differential operator BD,ε, 0 < ε ≤ 1, is considered with the Dirichlet boundary condition. The principal part of the operator is given in a factorized form. The operator involves first and zero order terms. The operator BD,ε is positive definite; its coefficients are periodic and depend on x/ε. The behavior of the operator exponential e -BD,εt, t > 0, is studied as ε → 0. Approximations for the exponential e -BD,εt are obtained in the operator norm on L2(O;n) and in the norm of operators acting from L2(O;n) to the Sobolev space H1(O;n). The results are applied to homogenization of solutions of the first initial boundary-value problem for parabolic systems.

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