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 $R^{\epsilon }=\{(x_1,x_2)\in {\mathbb{R}}^2|x_1\in (0,1),-\epsilon b(x_1)<x_2<\epsilon G(x_1,x_1/{\epsilon }^{\alpha })\}$ 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.

20 pages, 4 figures

References in corpus (1)

• Homogenization in a thin domain with an oscillatory boundary

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