This paper investigates the regularity of weak solutions to div-curl systems, with a particular emphasis on anisotropic coefficients characterized by low regularity. The problem of Hölder regularity for the div-curl system with a single anisotropic coefficient, posed by Yin in 2016, has remained open. We fully resolve this long-standing open problem, and our results extend to the case of two anisotropic coefficients as well. We prove that solutions are Hölder-continuous whenever the coefficients are Hölder-continuous. Furthermore, higher Hölder regularity of the coefficients yields a corresponding improvement in the solutions’ Hölder regularity. Our results settle Yin's open question and further extend the regularity theory to the case of two anisotropic coefficients. The method developed herein relies on classical tools from partial differential equations-including the Helmholtz decomposition, Campanato space estimates, and uniform ellipticity estimates-and can be further applied to a wider class of linear systems with low-regularity coefficients.

div-curl systems, Hölder regularity, anisotropic coefficients