Thin domains with doubly oscillatory boundary

TL;DR

This paper investigates the asymptotic behavior of the Laplace operator in a thin 2D domain with oscillatory boundaries of different periods, deriving a homogenized limit that accounts for the boundary oscillations' influence.

Contribution

It introduces a novel analysis of how differing boundary oscillation periods affect the homogenized limit of the Laplace operator in thin domains.

Findings

Derived the homogenized limit problem considering boundary oscillations

Showed the influence of oscillation period differences on the limit behavior

Provided asymptotic analysis for Laplace operator with Neumann conditions

Abstract

We consider a 2-dimensional thin domain with order of thickness {\epsilon} which presents oscillations of amplitude also {\epsilon} on both boundaries, top and bottom, but the period of the oscillations are of different order at the top and at the bottom. We study the behavior of the Laplace operator with Neumann boundary condition and obtain its asymptotic homogenized limit as the parameter {\epsilon} goes to 0. We are interested in understanding how this different oscillatory behavior at the boundary, influences the limit problem.