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

We consider the time-harmonic Maxwell equations posed in R3. We prove a priori bounds on the solution for L infinity coefficients euro and mu satisfying certain monotonicity properties, with these bounds valid for arbitrarily-large frequency, and explicit in the frequency and properties of euro and mu. The class of coefficients covered includes (i) certain euro and mu for which well-posedness of the time-harmonic Maxwell equations had not previously been proved, and (ii) scattering by a penetrable C0 star-shaped obstacle where euro 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.(c) 2023 The Author(s). Published by Elsevier Masson SAS. This is an open access article under the CC BY license (http://creativecommons .org /licenses /by /4 .0/).

Automated summary

In this article, we consider the time-harmonic Maxwell equations in R3. We establish a priori bounds on the solution for L infinity coefficients ε and μ that satisfy certain monotonicity properties. These bounds hold for arbitrarily large frequency and are explicitly determined by the frequency and properties of ε and μ. The class of coefficients covered includes cases where well-posedness of the time-harmonic Maxwell equations was previously unproven and cases involving scattering by a penetrable C0 star-shaped obstacle with smaller ε and μ inside the obstacle. The bounds obtained are uniform across all such obstacles and provide the first sharp frequency-explicit bounds for this problem at high-frequency.