Elliptic Equations in Weak Oscillatory Thin Domains: Beyond Periodicity with Boundary-Concentrated Reaction Terms

TL;DR

This paper investigates the asymptotic behavior of solutions to elliptic equations in weakly oscillating thin domains with boundary-concentrated reactions, extending classical periodic assumptions to quasiperiodic and almost-periodic cases.

Contribution

It introduces a new analysis framework for elliptic problems in non-periodic oscillatory thin domains, including boundary concentration effects, and proves convergence to a 1D limit model.

Findings

Solutions converge to a 1D limit equation capturing boundary oscillations.

The limit model accounts for quasiperiodic and almost-periodic boundary behaviors.

Numerical experiments validate the theoretical convergence results.

Abstract

In this paper we analyze the limit behavior of a family of solutions of the Laplace operator with homogeneous Neumann boundary conditions, set in a two-dimensional thin domain which presents weak oscillations on both boundaries and with terms concentrated in a narrow oscillating neighborhood of the top boundary. The aim of this problem is to study the behavior of the solutions as the thin domain presents oscillatory behaviors beyond the classical periodic assumptions,including scenarios like quasiperiodic or almost-periodic oscillations. We then prove that the family of solutions converges to the solution of a 1 dimensional limit equation capturing the geometry and oscillatory behavior of boundary of the domain and the narrow strip where the concentration terms take place. In addition, we include a series of numerical experiments illustrating the theoretical results obtained in the quasiperiodic context.