  In this paper, we study the regularity of the solutions of Maxwell's
equations in a bounded domain. We consider several different types of low
regularity assumptions to the coefficients which are all less than Lipschitz.
We first develop a new approach by giving $\mathcal{H}^1$ estimate when the
coefficients are $\mathcal{L}^{\infty}$ bounded; and then we derive
$\mathcal{W}^{1,p}$ estimates for every $p > 2$ when one of the leading
coefficients is simply continuous; Finally, we extend the result to
$\mathcal{C}^{1,\alpha}$ almost everywhere for the solution of the homogeneous
Maxwell's equations when the coefficients are $\mathcal{W}^{1,p}, \, p>3$ and
close to the identity matrix in the sense of $\mathcal{L}^{\infty}$ norm. The
last two estimates are new, and the techniques and methods developed here can
also be applied to other problems with similar difficulties.
