The Failure of Uniform Exponential Decay for Boundary Layers

Abstract

In the homogenization of elliptic boundary value problems with periodically oscillating coefficients, boundary layers are used to describe the boundary behavior of the solution. In the classical case, when the domain is the half space Ω = x ∈ ℝⁿ,x _n > 0, the boundary layers are defined on the semi-infinite strip ]0,l[ⁿ⁻¹x]0,∞[, and their energies decrease exponentially with respect to the second variable. In [11] we have shown that the property of uniform exponential decay of the boundary layers does not hold in general.

In our contribution, we improve the result of [11] by showing that for general domains the optimal decay of the energy of the boundary layers is polynomial of degree −1.