Homogenization of the Neumann problem for higher order elliptic equations with periodic coefficients

Abstract

Let be a bounded domain of class . In , we study a self-adjoint strongly elliptic operator of order 2p given by the expression , , with Neumann boundary conditions. Here, is a bounded and positive definite matrix-valued function in , periodic with respect to some lattice; is a differential operator of order p. The symbol is subject to some condition ensuring strong ellipticity of the operator . We find approximations for the resolvent in different operator norms with error estimates depending on and .

Keywords:

• Periodic differential operators
• higher order elliptic equations
• Neumann problem
• homogenization
• operator error estimates

Notes

No potential conflict of interest was reported by the author.

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Additional information

Funding

This work was supported by Russian Science Foundation [grant number 17-11-01069].