# Two-scale convergence in thin domains with locally periodic rapidly oscillating boundary

## TL;DR

This paper extends two-scale convergence to singular measures, analyzing the asymptotic behavior of elliptic operators in thin, locally periodic domains with oscillating boundaries, revealing new convergence properties and effective models.

## Contribution

It introduces a novel adaptation of two-scale convergence for singular measures and applies it to analyze elliptic operators in thin, oscillating domains with locally periodic structures.

## Key findings

- Convergence of measures to a 1D Lebesgue measure in thin domains
- Asymptotic analysis of elliptic operators with locally periodic coefficients
- Development of a new two-scale convergence framework for singular measures

## Abstract

The aim of this paper is to adapt the notion of two-scale convergence in $L^p$ to the case of a measure converging to a singular one. We present a specific case when a thin cylinder with locally periodic rapidly oscillating boundary shrinks to a segment, and the corresponding measure charging the cylinder converges to a one-dimensional Lebegues measure of an interval. The method is then applied to the asymptotic analysis of linear elliptic operators with locally periodic coefficients in a thin cylinder with locally periodic rapidly varying thickness.