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Inverse scattering problem for transparent obstacles

Published online by Cambridge University Press:  24 October 2008

Vesselin Petkov
Vesselin Petkov
Affiliation:
Bulgarian Academy of Science, Sofia
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Extract

Let K ⊂ ℝ3 be an open bounded strictly convex domain with smooth connected compact boundary ∂K. Set

We wish to study the filtered scattering amplitude, related to the transmission problem

Here w± are the limiting values of w on ∂Ω; from the Ω;± side, while ∂w±/∂n are the corresponding limiting values of the normal derivative ∂w?/∂n on ∂Ω;. The function α(x) ∈ C∞(K¯), called the index of refraction, has the property α(x) > 0 for x ∈ K¯.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 92 , Issue 2 , September 1982 , pp. 361 - 367
Copyright
Copyright © Cambridge Philosophical Society 1982

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References

REFERENCES

(1)Andersson, K. and Melrose, R.The propagation of singularities along gliding rays. Invent. Math. 41 (1977), 197–232.CrossRefGoogle Scholar
(2)Georgiev, V. Wave fronts of solutions to boundary problems for symmetric dissipative systems. Serdika, Bulg. Math. Publ. (to appear).Google Scholar
(3)Georgiev, V.Inverse scattering problems for symmetric strictly hyperbolic systems. C. R. Acad. Sci. Bulg. 35 (1982). (To appear.)Google Scholar
(4)Georgiev, V. High frequency asymptotics of the filtered scattering amplitude for dissipative hyperbolic systems. (In preparation.)Google Scholar
(5)Ivrii, V.Wave fronts of solutions to boundary problems for symmetric hyperbolic systems. I. Main theorem. Sibir. Mat. Z. 20, no. 4 (1979), 741–751. (In Russian.)Google Scholar
(6)Ivrii, V.Wave fronts of solutions to boundary problems for symmetric hyperbolic systems. II. Systems with characteristics of constant multiplicity. Sibir. Mat. Z. 20, no. 5 (1979), 1022–1037. (In Russian.)Google Scholar
(7)Ivrii, V.Wave fronts of solutions to boundary problems for a class of symmetric hyperbolic systems. Sibir. Mat. Z. 21, no. 4 (1980), 62–71. (In Russian.)Google Scholar
(8)Lax, P. and Phillips, R.Scattering Theory (Academic Press, 1967).Google Scholar
(9)Lax, P. and Phillips, R.The scattering of sound waves by an obstacle. Comm. Pure Appl. Math. 30 (1977), 195–233.CrossRefGoogle Scholar
(10)Majda, A. and Taylor, M.Inverse scattering problems for transparent obstacles, electro-magnetic waves and hyperbolic systems. Comm. Part. Diff. Equations 2 (1977), 395–438.CrossRefGoogle Scholar
(11)Melrose, R. and Sjöstrand, J.Singularities of boundary value problems: I. Comm. Pure Appl. Math. 31 (1978), 593–617.CrossRefGoogle Scholar
(12)Melrose, R. and Sjöstrand, J. Propagation de singularités pour des problèmes aux limites d'ordre 2. Séminaire Goulaouic-Schwartz 1977–1978, exposé no. xv.Google Scholar
(13)Nosmas, J.Parametrix de problème de transmission pour l'équation des ondes. J. Math. Pures Appl. 56 (1977), 423–435.Google Scholar
(14)Petkov, V.Propagation des singularités pour le problème de transmission et application au problème de la diffusion. C. R. Acad. Sci., Paris 290 (1980), 735–755.Google Scholar
(15)Petkov, V.Propagation of singularities and inverse scattering problem for transparent obstacles. J. Math. Pures Appl. 60 (1982), 65–90.Google Scholar
(16)Taylor, M.Reflection of singularities of solutions to systems of differential equations. Comm. Pure Appl. Math. 28 (1975), 457–478.CrossRefGoogle Scholar
(17)Taylor, M.Grazing rays and reflections of singularities of solutions to wave equations, Part II (Systems). Comm. Pure Appl. Math. 28 (1976), 463–481.Google Scholar