The Neumann problem in thin domains with very highly oscillatory boundaries

TL;DR

This paper investigates the asymptotic behavior of solutions to the Neumann problem in highly oscillatory thin domains, revealing how extreme boundary oscillations influence solution properties and domain geometry.

Contribution

It provides a detailed analysis of the Neumann problem in thin domains with highly oscillatory boundaries, including more complex geometries beyond simple graphs.

Findings

Oscillatory boundary behavior significantly affects solution limits.

High-frequency oscillations dominate the asymptotic analysis.

Results extend to complex, non-graph domain geometries.

Abstract

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