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

Théophile Chaumont-Frelet; Andrea Moiola; Euan Spence

doi:10.1016/j.matpur.2023.09.004

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

Théophile Chaumont-Frelet, Andrea Moiola, Euan Spence

Research output: Contribution to journal › Article › peer-review

Abstract

We consider the time-harmonic Maxwell equations posed in R³. We prove a priori bounds on the solution for L^∞ 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 C⁰star-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.

Original language	English
Pages (from-to)	183-218
Journal	Journal de Mathématiques Pures et Appliquées
Volume	179
Early online date	25 Sept 2023
DOIs	https://doi.org/10.1016/j.matpur.2023.09.004
Publication status	Published - 30 Nov 2023

Bibliographical note

Funding Information:
We thank Giovanni S. Alberti (University of Genoa), Ralf Hiptmair (ETH Zürich), Steven Johnson (MIT), Dirk Pauly (Universität Duisburg-Essen), Luca Rondi (Università di Pavia) and Michael Taylor (University of North Carolina at Chapel Hill) for useful discussions. AM acknowledges support from GNCS–INDAM, from PRIN project “NA_FROM-PDEs” and from MIUR through the “Dipartimenti di Eccellenza” Programme (2018–2022) – Dept. of Mathematics, University of Pavia. EAS acknowledges support from EPSRC grant EP/R005591/1.

Funding Information:
AM acknowledges support from GNCS–INDAM , from PRIN project “NA_FROM-PDEs” and from MIUR through the “Dipartimenti di Eccellenza” Programme (2018–2022) – Dept. of Mathematics, University of Pavia. EAS acknowledges support from EPSRC grant EP/R005591/1 .

Funding

We thank Giovanni S. Alberti (University of Genoa), Ralf Hiptmair (ETH Zürich), Steven Johnson (MIT), Dirk Pauly (Universität Duisburg-Essen), Luca Rondi (Università di Pavia) and Michael Taylor (University of North Carolina at Chapel Hill) for useful discussions. AM acknowledges support from GNCS–INDAM, from PRIN project “NA_FROM-PDEs” and from MIUR through the “Dipartimenti di Eccellenza” Programme (2018–2022) – Dept. of Mathematics, University of Pavia. EAS acknowledges support from EPSRC grant EP/R005591/1. AM acknowledges support from GNCS–INDAM , from PRIN project “NA_FROM-PDEs” and from MIUR through the “Dipartimenti di Eccellenza” Programme (2018–2022) – Dept. of Mathematics, University of Pavia. EAS acknowledges support from EPSRC grant EP/R005591/1 .

Keywords

Heterogeneous media
High frequency
Maxwell
Transmission problem
Wellposedness

ASJC Scopus subject areas

Applied Mathematics
General Mathematics

Access to Document

10.1016/j.matpur.2023.09.004Licence: CC BY

https://arxiv.org/abs/2301.07092

Cite this

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title = "Explicit bounds for the high-frequency time-harmonic Maxwell equations in heterogeneous media",

abstract = "We consider the time-harmonic Maxwell equations posed in R3. We prove a priori bounds on the solution for L∞ 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 C0 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 firstsharp frequency-explicit bounds for this problem at high-frequency.",

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author = "Th{\'e}ophile Chaumont-Frelet and Andrea Moiola and Euan Spence",

note = "Funding Information: We thank Giovanni S. Alberti (University of Genoa), Ralf Hiptmair (ETH Z{\"u}rich), Steven Johnson (MIT), Dirk Pauly (Universit{\"a}t Duisburg-Essen), Luca Rondi (Universit{\`a} di Pavia) and Michael Taylor (University of North Carolina at Chapel Hill) for useful discussions. AM acknowledges support from GNCS–INDAM, from PRIN project “NA_FROM-PDEs” and from MIUR through the “Dipartimenti di Eccellenza” Programme (2018–2022) – Dept. of Mathematics, University of Pavia. EAS acknowledges support from EPSRC grant EP/R005591/1. Funding Information: AM acknowledges support from GNCS–INDAM , from PRIN project “NA_FROM-PDEs” and from MIUR through the “Dipartimenti di Eccellenza” Programme (2018–2022) – Dept. of Mathematics, University of Pavia. EAS acknowledges support from EPSRC grant EP/R005591/1 . ",

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AB - We consider the time-harmonic Maxwell equations posed in R3. We prove a priori bounds on the solution for L∞ 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 C0 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 firstsharp frequency-explicit bounds for this problem at high-frequency.

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

Abstract

Bibliographical note

Funding

Keywords

ASJC Scopus subject areas

Access to Document

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At the interface between semiclassical analysis and numerical analysis of Wave propogation problems

Cite this

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

Abstract

Bibliographical note

Funding

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Projects

At the interface between semiclassical analysis and numerical analysis of Wave propogation problems

Cite this