# Homogenization of a locally periodic oscillating boundary

## TL;DR

This paper studies the homogenization process for Laplace equations in domains with oscillating boundaries, establishing weak and strong convergence results under natural regularity assumptions.

## Contribution

It provides new homogenization results for mixed boundary value problems with locally periodic oscillating boundaries, including weak and strong convergence analyses.

## Key findings

- Weak $L^2$ convergence of solutions and flows established
- Strong $L^2$ convergence of extensions analyzed
- Homogenization holds under natural regularity conditions

## Abstract

This paper deals with the homogenization of a mixed boundary value problem for the Laplace operator in a domain with locally periodic oscillating boundary. The Neumann condition is prescribed on the oscillating part of the boundary, and the Dirichlet condition on a separate part. It is shown that the homogenization result holds in the sense of weak $L^2$ convergence of the solutions and their flows, under natural hypothesis on the regularity of the domain. The strong $L^2$ convergence of average preserving extensions of the solutions and their flows is also considered.