Estimates in homogenization of higher-order elliptic operators

Abstract

A divergent-type elliptic operator of arbitrary even order 2m is studied. Coefficients of the operator are -periodic, is a small parameter. The resolvent equation is solvable in the Sobolev space of order m for any , provided the parameter is sufficiently large, , where the bound depends only on constants from ellipticity condition. The limit equation is of the same type but with constant coefficients, that is, . The limit operator can be considered here, for instance, in the sense of G-convergence. We prove that the resolvent approximates in operator -norm with the estimate , as . We find also the approximation of the resolvent in operator -norm. This is the sum , where is a correcting operator whose structure is given. We prove the estimate , as .

Keywords:

• Homogenization
• G-convergence
• operator-type estimates in homogenization
• higher-order elliptic equations
• corrector

Notes

No potential conflict of interest was reported by the author.

Additional information

Funding

The author was supported by Russian Science Foundation [grant number 14-11-00398].