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

Abstract

We consider the time-harmonic Maxwell equations posed in ${\mathbb{R}}^3$. We prove a priori bounds on the solution for $L^{\infty }$ coefficients $\epsilon$ and $\mu$ satisfying certain monotonicity properties, with these bounds valid for arbitrarily-large frequency, and explicit in the frequency and properties of $\epsilon$ and $\mu$. The class of coefficients covered includes (i) certain $\epsilon$ and $\mu$ 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 $\epsilon$ and $\mu$ 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.

References in corpus (5)

• Mosco convergence for H(curl) spaces, higher integrability for Maxwell's equations, and stability in direct and inverse EM scattering problems
• Domain decomposition preconditioning for the high-frequency time-harmonic Maxwell equations with absorption
• On uniqueness for time harmonic anisotropic Maxwell's equations with piecewise regular coefficients
• Uniform energy bound and Morawetz estimate for extreme components of spin fields in the exterior of a slowly rotating Kerr black hole I: Maxwell field
• The Helmholtz equation in random media: well-posedness and a priori bounds