  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 $R^\epsilon$ given by the small parameter
$\epsilon$. 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.

Analysis of PDEs