Geodesic
The Steklov problem in a half-plane: the dependence of eigenvalues on a piecewise-constant coefficient
Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 32, Tome 297 (2003), pp. 162-190 
N. G. Kuznetsov; O. V. Motygin. The Steklov problem in a half-plane: the dependence of eigenvalues on a piecewise-constant coefficient. Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 32, Tome 297 (2003), pp. 162-190. http://geodesic.mathdoc.fr/item/ZNSL_2003_297_a10/
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     title = {The {Steklov} problem in a half-plane: the dependence of eigenvalues on a~piecewise-constant coefficient},
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

The Steklov problem considered in the paper describes free two-dimensional oscillations of an ideal, incompressible, heavy fluid in a half-plane covered by a rigid dock with two symmetric gaps. Equivalent reduction of the problem to two spectral problems for integral operators allows us to find limits for all eigenfrequencies as the spacing between gaps tends to both zero and infinity. For the fundamental eigenfrequency and the corresponding eigenfunction two terms are found in the asymptotic expansion as the spacing tends to infinity. It is proved that all eigenvalues are simple for any value of the spacing.

[1] W. Stekloff, “Sur les problèmes fondamentaux de la physique mathematique”, Ann. Sci. École Normale Sup., 19 (1902), 455–490 | MR | Zbl

[2] C. Bandle, Isoperimetric Inequalities and Applications, Pitman, Boston, 1980 | MR | Zbl

[3] G. V. Rozenblyum, M. Z. Solomyak, M. A. Shubin, Differentsialnye uravneniya s chastnymi proizvodnymi – 7, Itogi nauki i tekhniki. Sovremennye problemy matematiki. Fundamentalnye napravleniya, 64, VINITI, Moskva, 1989 | MR | Zbl

[4] N. Kuznetsov, O. Motygin, “Sloshing problem in a half-plane covered by a dock with two gaps: monotonicity and asymptotics of eigenvalues”, C. R. Acad. Sci. Paris, Sèr. II, 329:11 (2001), 791–796

[5] D. W. Fox, J. R. Kuttler, “Sloshing frequencies”, Z. Angew. Math. Phys., 34 (1983), 668–696 | DOI | MR | Zbl

[6] S. A. Nazarov, B. A. Plamenevskii, Ellipticheskie zadachi v oblastyakh s kusochno gladkoi granitsei, Nauka, M., 1991

[7] A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integraly i ryady, Nauka, M., 1981 | MR | Zbl

[8] L. Lewin, Polylogarithms and Associated Functions, North Holland, Amsterdam, 1981 | MR | Zbl

[9] B. A. Troesch, “Proof of a conjecture on sloshing frequencies”, Z. Angew. Math. Phys., 25 (1974), 655–657 | DOI | MR | Zbl

[10] B. A. Troesch, H. R. Troesch, “A remark on the sloshing frequencies for a half-space”, Z. Angew. Math. Phys., 23 (1972), 703–711 | DOI | MR | Zbl

[11] V. S. Vladimirov, Uravneniya matematicheskoi fiziki, Nauka, M., 1967 | MR

[12] M. Sh. Birman, M. Z. Solomyak, Spektralnaya teoriya samosopryazhennykh operatorov v gilbertovom prostranstve, Izd. LGU, L., 1980 | MR

[13] A. M. J. Davis, “Waves in the presence of an infinite dock with gap”, J. Inst. Math. Appl., 6 (1970), 141–156 | DOI | Zbl

[14] T. Kato, Perturbation theory for linear operators, Springer-Verlag, Berlin, Heidelberg, New York, 1980

[15] J. W. Miles, “On the eigenvalue problem for fluid sloshing in a half-space”, Z. Angew. Math. Phys., 23 (1972), 861–868 | DOI | MR