  We study the limiting absorption principle and the well-posedness of Maxwell
equations with anisotropic sign-changing coefficients in the time-harmonic
domain. The starting point of the analysis is to obtain Cauchy problems
associated with two Maxwell systems using a change of variables. We then derive
a priori estimates for these Cauchy problems using two different approaches.
The Fourier approach involves the complementing conditions for the Cauchy
problems associated with two elliptic equations, which were studied in a
general setting by Agmon, Douglis, and Nirenberg. The variational approach
explores the variational structure of the Cauchy problems of the Maxwell
equations. As a result, we obtain general conditions on the coefficients for
which the limiting absorption principle and the well-posedness hold. Moreover,
these {\it new} conditions are of a local character and easy to check. Our work
is motivated by and provides general sufficient criteria for the stability of
electromagnetic fields in the context of negative-index metamaterials.
