### Abstract

Let O ⊃ ℝ^{d} be a bounded domain of class C^{1,1}. In L_{2}(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},ε^{t}, t > 0, is studied as ε → 0. Approximations for the exponential e ^{-BD},ε^{t} are obtained in the operator norm on L_{2}(O;^{n}) and in the norm of operators acting from L_{2}(O;^{n}) to the Sobolev space H^{1}(O;^{n}). The results are applied to homogenization of solutions of the first initial boundary-value problem for parabolic systems.

Original language | English |
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Pages (from-to) | 935-978 |

Number of pages | 44 |

Journal | St. Petersburg Mathematical Journal |

Volume | 29 |

Issue number | 6 |

DOIs | |

Publication status | Published - 1 Jan 2018 |

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### Scopus subject areas

- Analysis
- Algebra and Number Theory
- Applied Mathematics

### Cite this

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*St. Petersburg Mathematical Journal*, vol. 29, no. 6, pp. 935-978. https://doi.org/10.1090/spmj/1521

**Homogenization of the first initial boundary-value problem for parabolic systems : Operator error estimates.** / Meshkova, Yu M.; Suslina, T. A.

Research output

TY - JOUR

T1 - Homogenization of the first initial boundary-value problem for parabolic systems

T2 - Operator error estimates

AU - Meshkova, Yu M.

AU - Suslina, T. A.

PY - 2018/1/1

Y1 - 2018/1/1

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.

KW - Homogenization

KW - Operator error estimates

KW - Parabolic systems

KW - Periodic differential operators

UR - http://www.scopus.com/inward/record.url?scp=85054406968&partnerID=8YFLogxK

U2 - 10.1090/spmj/1521

DO - 10.1090/spmj/1521

M3 - Article

AN - SCOPUS:85054406968

VL - 29

SP - 935

EP - 978

JO - St. Petersburg Mathematical Journal

JF - St. Petersburg Mathematical Journal

SN - 1061-0022

IS - 6

ER -