The H\"{o}lder regularity of div-curl system with anisotropic coefficients

TL;DR

This paper proves the H"older regularity of solutions to the div-curl system with anisotropic coefficients, resolving an open problem and showing regularity improves with smoother coefficients.

Contribution

It addresses the open problem of H"older regularity for the div-curl system with anisotropic coefficients and extends results to cases with two such coefficients.

Findings

Established H"older regularity for solutions with one anisotropic coefficient.

Extended regularity results to systems with two anisotropic coefficients.

Showed regularity degree improves with higher coefficient regularity.

Abstract

This research examines the regularity of weak solutions to the Div-Curl system with low regularity anisotropic coefficients. The H\"older 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 H\"{o}lder regularity of the solution when the coefficients is H\"{o}lder continuous. Moreover, the degree of H\"{o}lder regularity of the solution can be improved if the coefficient has a greater degree of H\"{o}lder regularity.