Abstract
In this paper, the interior transmission problem for the non absorbing, anisotropic and inhomogeneous elasticity is investigated. The direct scattering problem for the penetrable inhomogeneous, anisotropic and nondissipative scatterer is first studied and the existence and uniqueness of its solution are established. In the sequel, the interior transmission problem in its classical and weak form is presented and suitable variational formulations of it are settled. Finally, it is proved that the interior transmission eigenvalues constitute a discrete set.
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D. Colton and A. Kirsch, A simple method for solving the inverse scattering problems in the resonance region. Inverse Problems 12 (1996) 383–393.
D. Colton, M. Piana and R. Potthast, A simple method using Morozov's discrepancy principle for solving scattering problems. Inverse Problems 13 (1999) 1477–1493.
D. Colton, J. Coyle and P. Monk, Recent developments in inverse acoustic scattering theory. SIAM Rev. 42(3) (2000) 369–414.
B.P. Rynne and B.D. Sleeman, The interior transmission problem and inverse scattering from inhomogeneous medium. SIAM J. Math. Anal. 22(6) (1991) 1755–1762.
A. Charalambopoulos, D. Gintides and K. Kiriaki, The linear sampling method for the transmission problem in three-dimensional linear elasticity. Inverse Problems 18 (2002) 547–558.
D. Colton, R. Kress and P. Monk, Inverse scattering from an orthotropic medium. J. Comput. Appl. Math. 81 (1997) 269–298.
D. Colton and R. Potthast, The inverse electromagnetic scattering problem for an anisotropic medium. Quart. J. Mech. Appl. Math. 52 (1999) 349–372.
J. Coyle, An inverse electromagnetic scattering problem in a two-layer background. Inverse Problems 16 (2000) 275–292.
P. Hahner, On the uniqueness of the shape of a penetrable, anisotropic obstacle. J. Comput. Appl. Math. 116 (2000) 167–180.
M. Piana, On the uniqueness for anisotropic inhomogeneous inverse scattering problems. Inverse Problems 14 (1998) 1565–1579.
F. Cakoni, D. Colton and H. Haddar, The linear sampling method for anisotropic media. J. Comput. Appl. Math. (2002), to appear.
F. Cakoni and H. Haddar, The linear sampling method for anisotropic media: Part II. Preprints 2001/26, MSRI Berkeley, California.
A. Charalambopoulos, D. Gintides and K. Kiriaki, The linear sampling method for nonabsorbing penetrable elastic bodies (2002) submitted for publication.
R. Leis, Initial Boundary Value Problems in Mathematical Sciences. Wiley, New York (1986).
G. Dassios, K. Kiriaki and D. Polyzos, Scattering theorems for complete dyadic fields. Internat. J. Engrg. Sci. 33 (1995) 269–277.
V.D. Kupradze, Three-Dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity. North-Holland, Amsterdam (1979).
V. Isakov, Inverse problems for partial differential equations. In: Applied Mathematical Sciences, Vol. 127. Springer, New York (1997).
D. Gilbarg and N.S. Trundinger, Elliptic Partial Differential Equations of Second Order. Springer, Berlin (1977).
D. Gintides and K. Kiriaki, The far field equations in linear elasticity - An inversion scheme. Z. Angew. Math. Mech. 81 (2001) 305–316.
D. Colton and R. Kress, Inverse acoustic and electromagnetic scattering theory. Applied Mathematical Sciences, Vol. 93. Springer, New York (1992).
G. Dassios and Z. Rigou, Elastic Herglotz functions, SIAM J. Appl. Math. 55(5) (1995) 1345–1361.
C.J.S. Alves and R. Kress, On the far field operator in elastic obstacle scattering. IMA J. Appl. Math. 67 (2002) 1–21.
W. Rudin, Functional Analysis. McGraw-Hill, New York (1973).
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Charalambopoulos, A. On the Interior Transmission Problem in Nondissipative, Inhomogeneous, Anisotropic Elasticity. Journal of Elasticity 67, 149–170 (2002). https://doi.org/10.1023/A:1023958030304
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DOI: https://doi.org/10.1023/A:1023958030304