Existence and stability of electromagnetic Stekloff eigenvalues with a trace class modification

Samuel Cogar

doi:10.3934/ipi.2021011

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2021, Volume 15, Issue 4: 723-744. Doi: 10.3934/ipi.2021011

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Existence and stability of electromagnetic Stekloff eigenvalues with a trace class modification

Samuel Cogar

Department of Mathematics, Rutgers University, New Brunswick, Piscataway, NJ 08854, USA

Received: May 31, 2020

Revised: September 30, 2020

Early access: January 2021

Published: August 2021

This material is based upon work supported by the Army Research Office through the National Defense Science and Engineering Graduate (NDSEG) Fellowship, 32 CFR 168a

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A recent area of interest is the development and study of eigenvalue problems arising in scattering theory that may provide potential target signatures for use in nondestructive testing of materials. We consider a generalization of the electromagnetic Stekloff eigenvalue problem that depends upon a smoothing parameter, for which we establish two main results that were previously unavailable for this type of eigenvalue problem. First, we use the theory of trace class operators to prove that infinitely many eigenvalues exist for a sufficiently high degree of smoothing, even for an absorbing medium. Second, we leverage regularity results for Maxwell's equations in order to establish stability results for the eigenvalues with respect to the material coefficients, and we show that this generalized class of Stekloff eigenvalues converges to the standard class as the smoothing parameter approaches zero.

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Mathematics Subject Classification: Primary: 35J25, 35P05, 35P25; Secondary: 35R30.

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