Large-scale boundary estimates of parabolic homogenization over rough boundaries

TL;DR

This paper develops large-scale boundary estimates for second-order parabolic systems with rapidly oscillating, time-dependent periodic coefficients over rough boundaries, using a two-step quantitative approximation approach.

Contribution

It introduces a novel quantitative method to approximate parabolic problems over rough boundaries by successive homogenization steps.

Findings

Established boundary estimates for rough boundary problems

Quantified the approximation errors in homogenization

Extended homogenization techniques to time-dependent coefficients

Abstract

In this paper, for a family of second-order parabolic system or equation with rapidly oscillating and time-dependent periodic coefficients over rough boundaries, we obtain the large-scale boundary estimates, by a quantitative approach. The quantitative approach relies on approximating twice: we first approximate the original parabolic problem over rough boundary by the same equation over a non-oscillating boundary and then approximate the oscillating equation over a non-oscillating boundary by its homogenized equation over the same non-oscillating boundary.