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August  2008, 2(3): 335-340. doi: 10.3934/ipi.2008.2.335

Resonances and balls in obstacle scattering with Neumann boundary conditions

1. 

Department of Mathematics, University of Missouri, Columbia, Missouri 65211, United States

Received  January 2008 Revised  June 2008 Published  July 2008

We consider scattering by a smooth obstacle in $R^d$, $d\geq 3 $ odd. We show that for the Neumann Laplacian if an obstacle has the same resonances as the ball of radius $\rho$ does, then the obstacle is a ball of radius $\rho$. We give related results for obstacles which are disjoint unions of several balls of the same radius.
Citation: T. J. Christiansen. Resonances and balls in obstacle scattering with Neumann boundary conditions. Inverse Problems & Imaging, 2008, 2 (3) : 335-340. doi: 10.3934/ipi.2008.2.335
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show all references

References:
[1]

Mat. Sbornik, 2 (1937), 1205-1238. Google Scholar

[2]

Classics of Soviet Mathematics, 4. Gordon and Breach Publishers, Amsterdam, 1996.  Google Scholar

[3]

Comm. Partial Differential Equations, 7 (1982), 905-958. doi: 10.1080/03605308208820241.  Google Scholar

[4]

Comm. Partial Differential Equations, 15 (1990), 245-272. doi: 10.1080/03605309908820686.  Google Scholar

[5]

Comm. Partial Differential Equations, 23 (1998), 933-948. doi: 10.1080/03605309808821373.  Google Scholar

[6]

Adv. in Math., 32 (1979), 204-232. doi: 10.1016/0001-8708(79)90042-2.  Google Scholar

[7]

J. Funct. Anal., 169 (1999), 604-609. doi: 10.1006/jfan.1999.3487.  Google Scholar

[8]

J. Funct. Anal., 45 (1982), 29-40. doi: 10.1016/0022-1236(82)90003-9.  Google Scholar

[9]

J. Funct. Anal., 53 (1983), 287-303. doi: 10.1016/0022-1236(83)90036-8.  Google Scholar

[10]

Journées Équations aux Dérivées partielles (1984), 1-8. Google Scholar

[11]

Cambridge University Press, Cambridge, 1995.  Google Scholar

[12]

V. Petkov and L. Stoyanov, "Geometry of Reflecting Rays and Inverse Spectral Problems,'', Pure and Applied Mathematics (New York). John Wiley & Sons, ().   Google Scholar

[13]

Ann. Henri Poincaré, 2 (2001), 675-711. doi: 10.1007/PL00001049.  Google Scholar

[14]

Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978.  Google Scholar

[15]

in "Partial Differential Equations and Mathematical Physics (Copenhagen, 1995; Lund, 1995),'' 264-285, Progr. Nonlinear Differential Equations Appl., 21, Birkhuser Boston, Boston, MA, 1996.  Google Scholar

[16]

J. Amer. Math. Soc., 4 (1991), 729-769.  Google Scholar

[17]

J. Funct. Anal., 123 (1994), 336-367. doi: 10.1006/jfan.1994.1092.  Google Scholar

[18]

Applied Mathematical Sciences, 116. Springer-Verlag, New York, 1996.  Google Scholar

[19]

Séminaire sur les Équations aux Dérivées Partielles, 1996-1997, Exp. No. XIII, 14pp., École Polytech., Palaiseau, 1997.  Google Scholar

[20]

Asian J. Math., 2 (1998), 609-617.  Google Scholar

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