Regularity of Homogenized Boundary Data in Periodic Homogenization of Elliptic Systems

TL;DR

This paper investigates the regularity of boundary data in periodic homogenization of elliptic systems, proving they belong to certain Sobolev spaces and are H"older continuous, which advances understanding of boundary layer behavior.

Contribution

It establishes that homogenized boundary data are in $W^{1,p}$ for all $1<p< obreak ext{ } obreakinity$, showing improved regularity results in elliptic homogenization.

Findings

Homogenized boundary data are in $W^{1,p}$ for all $1<p< obreak ext{ } obreakinity$

Boundary layer tails are H"older continuous of any order $eta ext{ in } (0,1)$

Regularity results apply to both Dirichlet and Neumann boundary conditions

Abstract

This paper is concerned with periodic homogenization of second-order elliptic systems in divergence form with oscillating Dirichlet data or Neumann data of first order. We prove that the homogenized boundary data belong to $W^{1, p}$ for any $1<p<\infty$. In particular, this implies that the boundary layer tails are H\"older continuous of order $\alpha$ for any $\alpha \in (0,1)$.