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Journal article
ASYMPTOTIC EXPANSIONS OF TRANSMISSION EIGENVALUES FOR SMALL PERTURBATIONS OF MEDIA WITH GENERALLY SIGNED CONTRAST
Inverse problems and imaging (Springfield, Mo.), v 12(4), pp 971-992
01 Aug 2018
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Abstract
In this paper we revisit the transmission eigenvalue problem for an inhomogeneous media of compact support perturbed by small penetrable homogeneous inclusions. Assuming that the inhomogeneous background media is known and smooth, we investigate how these small volume inclusions affect the transmission eigenvalues. Our perturbation analysis makes use of the formulation of the transmission eigenvalue problem introduced Kirsch in [8], which requires that the contrast of the inhomogeneity is of one-sign only near the boundary. Thus, our approach can handle small perturbations with positive, negative or zero (voids) contrasts. In addition to proving the convergence rate for the eigenvalues corresponding to the perturbed media as inclusions' volume goes to zero, we also provide the explicit first correction term in the asymptotic expansion for simple eigenvalues. The correction term involves computable information about the known inhomogeneity as well as the location, size and refractive index of small perturbations. Our asymptotic formula has the potential to be used to recover information about small inclusions from knowledge of the real transmission eigenvalues, which can be determined from scattering data.
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Details
- Title
- ASYMPTOTIC EXPANSIONS OF TRANSMISSION EIGENVALUES FOR SMALL PERTURBATIONS OF MEDIA WITH GENERALLY SIGNED CONTRAST
- Creators
- Fioralba Cakoni - Florida State UniversityShari Moskow - Drexel UniversityScott Rome - Drexel UniversityDepartment of Mathematics, Rutgers University, Piscataway, NJ 08854, USA
- Publication Details
- Inverse problems and imaging (Springfield, Mo.), v 12(4), pp 971-992
- Publisher
- Amer Inst Mathematical Sciences-Aims
- Number of pages
- 22
- Grant note
- FA9550-17-1-0147 / AFOSR; United States Department of Defense; Air Force Office of Scientific Research (AFOSR) DMS1602802; DMS1411721; DMS1715425 / NSF; National Science Foundation (NSF) 1602802 / Direct For Mathematical & Physical Scien; National Science Foundation (NSF); NSF - Directorate for Mathematical & Physical Sciences (MPS) 392261 / Simons Foundation
- Resource Type
- Journal article
- Language
- English
- Academic Unit
- Mathematics
- Web of Science ID
- WOS:000446987400007
- Scopus ID
- 2-s2.0-85051727359
- Other Identifier
- 991019167943704721
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- Collaboration types
- Domestic collaboration
- Web of Science research areas
- Mathematics, Applied
- Physics, Mathematical