The Neumann problem in thin domains with very highly oscillatory boundaries

In this paper we analyze the behavior of solutions of the Neumann problem posed in a thin domain of the type $R^\epsilon = {(x_1,x_2) \in \R^2 ; | ; x_1 \in (0,1), , - , \epsilon , b(x_1) < x_2 < \epsilon , G(x_1, x_1/\epsilon^\alpha) }$with$\alpha>1$and$\epsilon > 0$, defined by smooth functions $b(x)$ and $G(x,y)$, where the function $G$ is supposed to be $l(x)$-periodic in the second variable $y$. The condition $\alpha >1$ implies that the upper boundary of this thin domain presents a very high oscillatory behavior. Indeed, we have that the order of its oscillations is larger than the order of the amplitude and height of $R^{ϵ}$ given by the small parameter $ϵ$. We also consider more general and complicated geometries for thin domains which are not given as the graph of certain smooth functions, but rather more comb-like domains.