Effective laws for the Poisson equation on domains with curved oscillating boundaries

Abstract

In this article, we derive approximations and effective boundary laws for solutions u ^∊ of the Poisson equation on a domain whose boundary differs from the smooth boundary of a domain by rapid oscillations of size ε. We construct a boundary layer correction which yields an O(ε) approximation in the energy norm and an approximation in the L ²-norm for a right-hand side . We also show that the same approximation order can be obtained for the error on subdomains by solving an effective equation on Ω satisfying a boundary condition of Robin type.

Keywords:

• Homogenization
• Poisson equation
• Oscillating boundary
• Curved boundary
• Robin boundary condition
• Boundary layer
• Unbounded domain
• Mathematical Subject Classifications:
• 35B27
• 35J05
• 35J25

Notes

†Note that for c ^bl >0, problem (99) can be ill-posed without further restictions on ∊ and c^bl .