Homogenization in a thin domain with an oscillatory boundary

TL;DR

This paper studies the asymptotic behavior of the Laplace operator with Neumann boundary conditions in a thin, oscillatory domain, deriving homogenized limits as the domain's thickness and boundary oscillations vanish.

Contribution

It provides a rigorous homogenization analysis for a thin domain with highly oscillatory boundary, capturing the combined effects of domain thickness and boundary oscillations.

Findings

Derived effective limit equations for the Laplace operator in the thin oscillatory domain.

Identified the influence of boundary oscillations on the homogenized problem.

Established convergence results as the domain parameter epsilon tends to zero.

Abstract

In this paper we analyze the behavior of the Laplace operator with Neumann boundary conditions in a thin domain of the type $R^\epsilon = \{(x,y) \in \R^2; x \in (0,1), 0 < y < \epsilon G(x, x/\epsilon)\} $ where the function G(x,y) is periodic in y of period L. Observe that the upper boundary of the thin domain presents a highly oscillatory behavior and, moreover, the height of the thin domain, the amplitude and period of the oscillations are all of the same order, given by the small parameter $\epsilon$.