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 ϵ = { ( x 1 , x 2 ) ∈ R 2 ∣ x 1 ∈ ( 0 , 1 ) , − ϵ b ( x 1 ) < x 2 < ϵ G ( x 1 , x 1 / ϵ α ) } with α > 1 and ϵ > 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 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.