The Neumann problem in thin domains with very highly oscillatory boundaries

Abstract

In this paper we analyze the behavior of solutions of the Neumann problem posed in a thin domain of the type $R^ε= \{(x_1,x_2) \in \R^2 \; | \; x_1 \in (0,1), \, - \, ε\, b(x_1) < x_2 < ε\, G(x_1, x_1/ε^α) \}$ 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.