On Lp Estimates in Homogenization of Elliptic Equations of Maxwell's Type

TL;DR

This paper establishes uniform Lp estimates for solutions of Maxwell-type elliptic systems with rapidly oscillating periodic coefficients in smooth domains, advancing the understanding of homogenization in electromagnetic problems.

Contribution

It provides the first uniform Lp and curl estimates for Maxwell's systems with periodic coefficients in $C^{1, \alpha}$ domains, extending scalar elliptic results to vector systems.

Findings

Uniform Lp estimates for solutions and their curls

Extension of scalar elliptic estimates to Maxwell systems

Applicable to domains with $C^{1, \alpha}$ regularity

Abstract

For a family of second-order elliptic systems of Maxwell's type with rapidly oscillating periodic coefficients in a $C^{1, \alpha}$ domain $\Omega$, we establish uniform estimates of solutions $u_\varep$ and $\nabla \times u_\varep$ in $L^p(\Omega)$ for $1<p\le \infty$. The proof relies on the uniform $W^{1,p}$ and Lipschitz estimates for solutions of scalar elliptic equations with periodic coefficients.