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

Théophile Chaumont-Frelet, Andrea Moiola, Euan Spence

Research output: Contribution to journalArticlepeer-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 Cstar-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 languageEnglish
Pages (from-to)183-218
JournalJournal de Mathématiques Pures et Appliquées
Volume179
Early online date25 Sept 2023
DOIs
Publication statusPublished - 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 .

Keywords

  • Heterogeneous media
  • High frequency
  • Maxwell
  • Transmission problem
  • Wellposedness

ASJC Scopus subject areas

  • Applied Mathematics
  • Mathematics(all)

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