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首页> 外文期刊>St. Petersburg mathematical journal >HOMOGENIZATION OF ELLIPTIC SYSTEMS WITH PERIODIC COEFFICIENTS: OPERATOR ERROR ESTIMATES IN L-2(R-d) WITH CORRECTOR TAKEN INTO ACCOUNT
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HOMOGENIZATION OF ELLIPTIC SYSTEMS WITH PERIODIC COEFFICIENTS: OPERATOR ERROR ESTIMATES IN L-2(R-d) WITH CORRECTOR TAKEN INTO ACCOUNT

机译：具有周期系数的椭圆系统的均质化：考虑到帐务员的纠正，L-2（R-d）中的算子误差估计

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摘要

A matrix elliptic selfadjoint second order differential operator (DO) B-epsilon with rapidly oscillating coefficients is considered in L-2(R-d; C-n). The principal part b(D)* g(epsilon(-1)x) b(D) of this operator is given in a factorized form, where g is a periodic, bounded, and positive definite matrix-valued function and b(D) is a matrix first order DO whose symbol is a matrix of maximal rank. The operator B-epsilon also includes first and zero order terms with unbounded coefficients. The problem of homogenization in the small period limit is studied. For the generalized resolvent of B-epsilon, approximation in the L-2(R-d; C-n)-operator norm with an error O(epsilon(2)) is obtained. The principal term of this approximation is given by the generalized resolvent of the effective operator B-0 with constant coefficients. The first order corrector is taken into account. The error estimate obtained is order sharp; the constants in estimates are controlled in terms of the problem data. General results are applied to homogenization problems for the Schrodinger operator and the two-dimensional Pauli operator with singular rapidly oscillating potentials.
机译：在L-2（R-d; C-n）中考虑具有快速振荡系数的矩阵椭圆自伴二阶微分算子（DO）B-ε。该算子的主要部分b（D）* g（epsilon（-1）x）b（D）以因子分解形式给出，其中g是周期，有界和正定矩阵值函数，b（D ）是矩阵一阶DO，其符号是最大秩矩阵。算子B-ε还包括具有无界系数的一阶和零阶项。研究了小周期限制下的均质化问题。对于B-ε的广义分解物，可以得到L-2（R-d; C-n）-算子范数的近似值，其误差为O（epsilon（2））。该近似值的主要术语由有效算子B-0的常数为常数的广义解析度给出。考虑一阶校正器。所获得的误差估计为锐度;估计中的常数根据问题数据进行控制。将一般结果应用于奇异快速振荡势的Schrodinger算符和二维Pauli算子的均化问题。

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