Homogenization of a mixed boundary condition

Abstract

We study the asymptotic behavior of the periodically mixed boundary value problem. The Dirichlet and Neumann boundary conditions are non-homogeneous and periodically mixed with small period ε. Using asymptotic analysis with respect to $\epsilon \ll 1$, we derive an asymptotic approximation that has boundary condition of the Robin type. We justify the obtained condition by proving an appropriate error estimate.

Keywords:

• Periodic boundary condition
• homogenization
• boundary layer
• Laplace equation
• the Zaremba problem

AMS subject classifications:

• 35B27
• 35B25
• 35B40

Acknowledgments

This paper is dedicated to our friend, colleague and teacher Andro Mikelić.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work has been supported by the Croatian Science Foundation [project AsAn IP-2018-01-2735].