Thin domains with extremely high oscillatory boundaries

Abstract. In this paper we analyze the behavior of solutions of the Neumann problem posed in a thin domain of the type R = {(x1, x2) ∈ R2 | x1 ∈ (0, 1), − b(x1) < x2 < G(x1, x1/α)} with α> 1 and > 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 α> 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 given by the small parameter . 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. 1