Karp's Theorem in Inverse Obstacle Scattering Problems

  • Jaemin Shin Hanbat National University
Keywords: Inverse obstacle scattering problem, Karp's Theorem, Symmetric problem

Abstract

In this work, we provide a proof of the so-called Karp's theorem in a different approach. We use the unique continuation principle together with the monotonicity of eigenvalues for the negative Laplace operator. This method is new and would be applicable to other types of inverse scattering problems.

References

David Colton and Rainer Kress. Inverse acoustic and electromagnetic scattering theory, volume 93 of Applied

Mathematical Sciences. Springer, New York, third edition, 2013.

Alexander G. Ramm. Scattering by obstacles, volume 21 of Mathematics and its Applications. D. Reidel Publishing

Co., Dordrecht, 1986.

David Colton and Rainer Kress. Looking back on inverse scattering theory. SIAM Rev., 60(4):779{807, 2018.

Samuel N. Karp. Far eld amplitudes and inverse diraction theory. In Electromagnetic waves, pages 291{300.

Univ. of Wisconsin Press, Madison, Wis., 1962.

David Colton and Andreas Kirsch. Karp's theorem in acoustic scattering theory. Proc. Amer. Math. Soc.,

(3):783{788, 1988.

David Colton and Rainer Kress. Karp's theorem in electromagnetic scattering theory. Proc. Amer. Math. Soc.,

(3):764{769, 1988.

P. A. Martin and G. Dassios. Karp's theorem in elastodynamic inverse scattering. Inverse Problems, 9(1):97{111,

Alexander G. Ramm. Symmetry properties of scattering amplitudes and applications to inverse problems. J.

Math. Anal. Appl., 156(2):333{340, 1991.

Alexander G. Ramm. Symmetry problem. Proc. Amer. Math. Soc., 141(2):515{521, 2013.

Alexander G. Ramm. Symmetry problems for the Helmholtz equation. Appl. Math. Lett., 96:122{125, 2019.

Fioralba Cakoni and David Colton. A Qualitative Approach to Inverse Scattering Theory. Applied Mathematical

Sciences. Springer US, 2013.

Rolf Leis. Initial boundary value problems in mathematical physics. Teubner, 1986.

L. E. Payne and H. F. Weinberger. An optimal Poincare inequality for convex domains. Arch. Rational Mech.

Anal., 5:286{292 (1960), 1960.

Published
2019-08-14
How to Cite
Shin, J. (2019). Karp’s Theorem in Inverse Obstacle Scattering Problems. JOURNAL OF ADVANCES IN MATHEMATICS, 17, 34-38. https://doi.org/10.24297/jam.v17i0.8399
Section
Articles