The Hölder regularity of div-curl system with anisotropic coefficients

TL;DR

This work establishes Hölder regularity for weak solutions of the div-curl system with anisotropic coefficients in $\Omega\subset\mathbb{R}^3$, first handling the partial anisotropy case and then extending to full anisotropy by transforming $u$ to $\tilde{u}=Bu$. The authors employ a Helmholtz decomposition $u=\nabla q+\operatorname{curl}\Phi$ and Campanato-space techniques to derive $C^{\alpha}$ bounds when $A,B\in C^{\alpha}$ and the data belong to $L^{p}$ or Campanato-type spaces, with $\alpha=(\tau-1)/2$ for $\tau\in(1,3)$. They further show higher-order regularity: if $A,B\in C^{k,\alpha}$ and $f,g\in C^{k-1,\alpha}$, then $u\in C^{k,\alpha}$, and provide a Hölder estimate framework applicable to Maxwell-type curl-curl systems. As an application, Hölder regularity is obtained for the curl of the magnetic field $H$ in time-harmonic Maxwell equations when permittivity $\varepsilon$ is Hölder continuous, illustrating the practical impact for electromagnetic theory.

Abstract

This research examines the regularity of weak solutions to the Div-Curl system with low regularity anisotropic coefficients. The Hölder regularity of the Div-Curl system with one anisotropic coefficient was an unresolved problem raised by Yin in 2016. We have addressed the open problem, and the findings extend to the scenario involving two anisotropic coefficients. We establish the Hölder regularity of the solution when the coefficients is Hölder continuous. Moreover, the degree of Hölder regularity of the solution can be improved if the coefficient has a greater degree of Hölder regularity.

Theorems & Definitions (21)

• Lemma 1: Helmholtz decomposition
• Theorem 1
• proof
• Remark 1
• Theorem 2
• Remark 2
• Theorem 3
• proof
• Theorem 4
• proof