Explicit bounds for the high-frequency time-harmonic Maxwell equations in heterogeneous media

We consider the time-harmonic Maxwell equations posed in $R^3$. We prove a priori bounds on the solution for $L^{\infty }$ coefficients ϵ and μ satisfying certain monotonicity properties, with these bounds valid for arbitrarily-large frequency, and explicit in the frequency and properties of ϵ and μ. The class of coefficients covered includes (i) certain ϵ and μ for which well-posedness of the time-harmonic Maxwell equations had not previously been proved, and (ii) scattering by a penetrable $C^0$ star-shaped obstacle where ϵ and μ are smaller inside the obstacle than outside. In this latter setting, the bounds are uniform across all such obstacles, and the first sharp frequency-explicit bounds for this problem at high-frequency.