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Journal Articles Journal de Mathématiques Pures et Appliquées Year : 2023

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.
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hal-04001866 , version 1 (23-02-2023)

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Théophile Chaumont-Frelet, Andrea Moiola, Euan A. Spence. Explicit bounds for the high-frequency time-harmonic Maxwell equations in heterogeneous media. Journal de Mathématiques Pures et Appliquées, 2023, 179, pp.183-218. ⟨10.1016/j.matpur.2023.09.004⟩. ⟨hal-04001866⟩
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