Homogenization in a thin domain with an oscillatory boundary

In this paper we analyze the behavior of the Laplace operator with Neumann boundary conditions in a thin domain of the type $R^\epsilon = {(x,y) \in \R^2; x \in (0,1), 0 < y < \epsilon G(x, x/\epsilon)} $ where the function G(x,y) is periodic in y of period L. Observe that the upper boundary of the thin domain presents a highly oscillatory behavior and, moreover, the height of the thin domain, the amplitude and period of the oscillations are all of the same order, given by the small parameter $ϵ$.