An $H^{s,p}(\curl;\Omega)$ estimate for the Maxwell system

TL;DR

This paper establishes an $H^{s,p}( ext{curl})$ estimate for Maxwell equations with anisotropic coefficients, extending known elliptic estimates to the Maxwell system under certain regularity conditions.

Contribution

It generalizes Gr{"o}ger's $L^p$ estimate to the Maxwell system, providing new regularity results for solutions with anisotropic coefficients.

Findings

Derived $H^{s,p}( ext{curl})$ estimates for Maxwell solutions.

Extended elliptic $L^p$ estimates to Maxwell equations.

Applicable to anisotropic media with specific regularity bounds.

Abstract

We derive an $H_{0}^{s,p}(\curl;\Omega)$ estimate for the solutions of the Maxwell type equations modeled with anisotropic and $W^{s, \infty}(\Omega)$-regular coefficients. Here, we obtain the regularity of the solutions for the integrability and smoothness indices $(p, s)$ in a plan domain characterized by the apriori lower/upper bounds of $a$ and the apriori upper bound of its H{\"o}lder semi-norm of order $s$. The proof relies on a perturbation argument generalizing Gr{\"o}ger's $L^p$-type estimate, known for the elliptic problems, to the Maxwell system.