Applications of Fourier analysis in homogenization of Dirichlet problem III: Polygonal Domains

TL;DR

This paper establishes convergence results for the homogenization of the Dirichlet problem with oscillating boundary data in convex polygonal domains, using integral representations and Diophantine conditions.

Contribution

It provides new pointwise and $L^p$ convergence results for polygonal domains under Diophantine conditions, extending homogenization theory.

Findings

Proves pointwise convergence of solutions.

Establishes $L^p$ convergence with near-optimal rates.

Identifies conditions for convergence in polygonal domains.

Abstract

In this paper we prove convergence results for the homogenization of the Dirichlet problem with rapidly oscillating boundary data in convex polygonal domains. Our analysis is based on integral representation of solutions. Under a certain Diophantine condition on the boundary of the domain and smooth coefficients we prove pointwise, as well as $L^p$ convergence results. For larger exponents $p$ we prove that the $L^p$ convergence rate is close to optimal. We shall also suggest several directions of possible generalization of the result in this paper.