Nature of the transmission eigenvalue spectrum for elastic bodies

Cédric Bellis, Fioralba Cakoni, Bojan B. Guzina

Research output: Contribution to journalArticlepeer-review

25 Scopus citations

Abstract

This study develops a spectral theory of the interior transmission problem (ITP) for heterogeneous and anisotropic elastic solids. The subject is central to the so-called qualitative methods for inverse scattering involving penetrable obstacles. Although simply stated as a coupled pair of elastodynamic wave equations, the ITP for elastic bodies is neither self-adjoint nor elliptic. To help deal with such impediments, earlier studies have established the well-posedness of an elastodynamic ITP under notably restrictive assumptions on the contrast in elastic and mass density parameters between the scatterer and the background solid. Due to lack of self-adjointness of the problem, these analyses were further successful in substantiating the discreteness of the relevant eigenvalue spectrum but not its existence. The aim of this work is to provide a systematic treatment of the ITP for elastic bodies that transcends the limitations of earlier analyses. Considering a broad range of material-contrast configurations, this paper investigates the questions of the solvability of the ITP, the discreteness of its eigenvalues and, for the first time, of the existence of such eigenvalue spectrum. Necessitated by the breadth of material configurations studied, the relevant claims are established via a suite of variational formulations, each customized to meet the needs of a particular subclass of eigenvalue problems.

Original languageEnglish (US)
Pages (from-to)895-923
Number of pages29
JournalIMA Journal of Applied Mathematics (Institute of Mathematics and Its Applications)
Volume78
Issue number5
DOIs
StatePublished - Oct 2013

Bibliographical note

Funding Information:
The support provided by the University of Minnesota Supercomputing Institute, the endowed Shimizu Professorship, and the U.S. Air Force Office of Scientific Research to F. Cakoni (FA9550-08-1-0138) is kindly acknowledged. Special thanks are extended to MTS Systems Corporation for providing the opportunity for F. Cakoni to visit the Department of Civil Engineering, University of Minnesota as an MTS Visiting Professor of Geomechanics.

Keywords

  • Inhomogeneous elastic media
  • Interior transmission problem
  • Inverse elastic scattering
  • Transmission eigenvalues

Fingerprint

Dive into the research topics of 'Nature of the transmission eigenvalue spectrum for elastic bodies'. Together they form a unique fingerprint.

Cite this