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 Rϵ={(x1,x2)∈R2∣x1∈(0,1),−ϵb(x1)1α>1 and ϵ>0ϵ>0, defined by smooth functions b(x)b(x) and G(x,y)G(x,y), where the function GG is supposed to be l(x)l(x)-periodic in the second variable yy. The condition α>1α>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 RϵRϵ given by the small parameter ϵϵ. 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.