Asymptotic Analysis of Laplacian Operator in Thin Domains on the Sphere with Highly Oscillatory Boundary

TL;DR

This paper studies the asymptotic behavior of solutions to the Poisson equation with Neumann boundary conditions in thin, oscillatory domains on the sphere, deriving homogenized limits and convergence rates.

Contribution

It introduces a homogenization framework for the Laplacian in thin oscillatory spherical domains, providing strong convergence results and error estimates.

Findings

Homogenized limit problem derived using Multiple Scales method

Strong convergence of solutions established with correctors

Error estimates quantified for the convergence process

Abstract

In this work we analyse the convergence of solutions of the Poisson equation with Neumann boundary conditions in a thin domain with highly oscillatory behavior $\mathcal{U}^\varepsilon$ contained in the sphere $\mathbb{S}^2$. Using the Multiple Scales method, we obtain the homogenized limit problem and analyse the convergence of solutions, as $\varepsilon$ tends to $0$. Introducing appropriate correctors, we show strong convergence and give error estimates.