Asymptotic expansions for eigenvalues of the Steklov problem in singularly perturbed domains

Nazarov, S.

doi:10.1090/S1061-0022-2015-01339-3

Asymptotic expansions for eigenvalues of the Steklov problem in singularly perturbed domains
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by S. A. Nazarov
Translated by: A. Plotkin

St. Petersburg Math. J. 26 (2015), 273-318

DOI: https://doi.org/10.1090/S1061-0022-2015-01339-3

Published electronically: February 3, 2015

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Abstract:

Full asymptotic expansions are constructed and justified for two series of eigenvalues and the corresponding eigenfunctions of the spectral Steklov problem in a domain with a singular boundary perturbation having the form of a small cavity. The terms of those series are of type $\lambda _k+o(1)$ and $\varepsilon ^{-1}(\mu _m+o(1))$, where $\lambda _k$ and $\mu _m$ are the eigenvalues of the Steklov problem in a bounded domain without cavity and the exterior Steklov problem for a cavity of unit size. A similar problem of the surface wave is also treated. The smoothness requirements on the boundary are discussed and unsolved problems are stated.

References

W. G. Mazja, S. A. Nasarow, and B. A. Plamenewski, Asymptotische Theorie elliptischer Randwertaufgaben in singulär gestörten Gebieten. I, Mathematische Lehrbücher und Monographien, II. Abteilung: Mathematische Monographien [Mathematical Textbooks and Monographs, Part II: Mathematical Monographs], vol. 82, Akademie-Verlag, Berlin, 1991 (German). Störungen isolierter Randsingularitäten. [Perturbations of isolated boundary singularities]. MR 1101139
Serhii Gryshchuk and Massimo Lanza de Cristoforis, Simple eigenvalues for the Steklov problem in a domain with a small hole. A functional analytic approach, Math. Methods Appl. Sci. 37 (2014), no. 12, 1755–1771. MR 3231071, DOI 10.1002/mma.2933
Massimo Lanza de Cristoforis, Asymptotic behavior of the solutions of the Dirichlet problem for the Laplace operator in a domain with a small hole. A functional analytic approach, Analysis (Munich) 28 (2008), no. 1, 63–93. MR 2396403, DOI 10.1524/anly.2008.0903
Massimo Lanza de Cristoforis, Simple Neumann eigenvalues for the Laplace operator in a domain with a small hole. A functional analytic approach, Rev. Mat. Complut. 25 (2012), no. 2, 369–412. MR 2931418, DOI 10.1007/s13163-011-0081-8
M. Lobo and M. Pérez, Local problems for vibrating systems with concentrated masses: a review, C. R. Mecanique 331 (2003), 303–317.
S. A. Nazarov, Asymptotic theory of thin plates and rads. Vol. 1, Dimension reduction and integral estimates, Nauch. Kniga, Novosibirsk, 2002. (Russian)
S. A. Nazarov, Uniform estimates of remainders in asymptotic expansions of solutions to the problem on eigenoscillations of a piezoelectric plate, J. Math. Sci. (N.Y.) 114 (2003), no. 5, 1657–1725. Function theory and applications. MR 1981301, DOI 10.1023/A:1022364812273
Miguel Lobo, Serguei A. Nazarov, and Eugenia Perez, Eigen-oscillations of contrasting non-homogeneous elastic bodies: asymptotic and uniform estimates for eigenvalues, IMA J. Appl. Math. 70 (2005), no. 3, 419–458. MR 2144635, DOI 10.1093/imamat/hxh039
S. A. Nazarov, Asymptotic behavior of the eigenvalues of the Steklov problem on a junction of domains of different limiting dimensions, Zh. Vychisl. Mat. Mat. Fiz. 52 (2012), no. 11, 2033–2049 (Russian, with Russian summary); English transl., Comput. Math. Math. Phys. 52 (2012), no. 11, 1574–1589. MR 3247705, DOI 10.1134/S0965542512110097
Fritz John, On the motion of floating bodies. I, Comm. Pure Appl. Math. 2 (1949), 13–57. MR 32328, DOI 10.1002/cpa.3160020102
Anne-Sophie Bonnet-Ben Dhia and Patrick Joly, Mathematical analysis of guided water waves, SIAM J. Appl. Math. 53 (1993), no. 6, 1507–1550. MR 1247167, DOI 10.1137/0153071
C. M. Linton and P. McIver, Embedded trapped modes in water waves and acoustics, Wave Motion 45 (2007), no. 1-2, 16–29. MR 2441664, DOI 10.1016/j.wavemoti.2007.04.009
N. Kuznetsov, V. Maz’ya, and B. Vainberg, Linear water waves, Cambridge University Press, Cambridge, 2002. A mathematical approach. MR 1925354, DOI 10.1017/CBO9780511546778
Sergey A. Nazarov and Jari Taskinen, On essential and continuous spectra of the linearized water-wave problem in a finite pond, Math. Scand. 106 (2010), no. 1, 141–160. MR 2603466, DOI 10.7146/math.scand.a-15129
Sergey A. Nazarov and Jari Taskinen, Radiation conditions at the top of a rotational cusp in the theory of water-waves, ESAIM Math. Model. Numer. Anal. 45 (2011), no. 5, 947–979. MR 2817552, DOI 10.1051/m2an/2011004
S. A. Nazarov, Concentration of trapped modes in problems of the linear theory of waves on the surface of a fluid, Mat. Sb. 199 (2008), no. 12, 53–78 (Russian, with Russian summary); English transl., Sb. Math. 199 (2008), no. 11-12, 1783–1807. MR 2489688, DOI 10.1070/SM2008v199n12ABEH003981
V. G. Maz′ya, S. A. Nazarov, and B. A. Plamenevskiĭ, Asymptotic expansions of eigenvalues of boundary value problems for the Laplace operator in domains with small openings, Izv. Akad. Nauk SSSR Ser. Mat. 48 (1984), no. 2, 347–371 (Russian). MR 740795
I. V. Kamotskiĭ and S. A. Nazarov, Spectral problems in singularly perturbed domains and selfadjoint extensions of differential operators [1 766 021], Proceedings of the St. Petersburg Mathematical Society, Vol. 6 (Russian), Tr. St.-Peterbg. Mat. Obshch., vol. 6, Nauchn. Kniga, Novosibirsk, 1998, pp. 151–212 (Russian). MR 1768332
G. Pólya and G. Szegö, Isoperimetric Inequalities in Mathematical Physics, Annals of Mathematics Studies, No. 27, Princeton University Press, Princeton, N. J., 1951. MR 0043486
O. A. Ladyzhenskaya, Kraevye zadachi matematicheskoĭ fiziki, Izdat. “Nauka”, Moscow, 1973 (Russian). MR 0599579
M. Š. Birman and M. Z. Solomjak, Spektral′naya teoriya samosopryazhennykh operatorov v gil′bertovom prostranstve, Leningrad. Univ., Leningrad, 1980 (Russian). MR 609148
M. I. Višik and L. A. Lyusternik, Regular degeneration and boundary layer for linear differential equations with small parameter, Uspehi Mat. Nauk (N.S.) 12 (1957), no. 5(77), 3–122 (Russian). MR 0096041
Fritz John, Plane waves and spherical means applied to partial differential equations, Interscience Publishers, New York-London, 1955. MR 0075429
L. V. Kantorovič and V. I. Krylov, Priblizhennye metody vysshego analiza, Fifth corrected edition, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow-Leningrad, 1962 (Russian). MR 0154389
V. A. Kondrat′ev, The smoothness of the solution of the Dirichlet problem for second order elliptic equations in a piecewise smooth domain, Differencial′nye Uravnenija 6 (1970), 1831–1843 (Russian). MR 0282052
V. G. Maz′ja and B. A. Plamenevskiĭ, Elliptic boundary value problems in a domain with a piecewise smooth boundary, Proceedings of the Symposium on Continuum Mechanics and Related Problems of Analysis (Tbilisi, 1971) Izdat. “Mecniereba”, Tbilisi, 1973, pp. 171–181 (Russian). MR 0412597
V. G. Maz′ja and B. A. Plamenevskiĭ, Schauder estimates for the solutions of elliptic boundary value problems in domains with edges on the boundary, Partial differential equations (Russian), Trudy Sem. S. L. Soboleva, No. 2, vol. 1978, Akad. Nauk SSSR Sibirsk. Otdel., Inst. Mat., Novosibirsk, 1978, pp. 69–102, 143 (Russian). MR 568035
S. A. Nazarov and B. A. Plamenevskiĭ, The Neumann problem for selfadjoint elliptic systems in a domain with a piecewise-smooth boundary, Trudy Leningrad. Mat. Obshch. 1 (1990), 174–211, 246–247 (Russian). MR 1104210
—, Elliptic problems in domains with piecewise smooth boundaries, Nauka, Moscow, 1991. (Russian)
S. A. Nazarov, Derivation of the variational inequality for the shape of small increment of the tensile crack, Izv. Ross. Akad. Nauk Meh. Tv. Tel. 2 (1989), 152–160. (Russian)
Alain Campbell and Sergueï A. Nazarov, Asymptotics of eigenvalues of a plate with small clamped zone, Positivity 5 (2001), no. 3, 275–295. MR 1836750, DOI 10.1023/A:1011469822255
M. A. Krasnosel′skiĭ, G. M. Vainikko, P. P. Zabreĭko, Ja. B. Rutickiĭ, and V. Ja. Stecenko, Priblizhennoe reshenie operatornykh uravneniĭ, Izdat. “Nauka”, Moscow, 1969 (Russian). MR 0259635
S. A. Nazarov, The Vishik-Lyusternik method for elliptic boundary value problems in regions with conic points. I. Problem in a cone, Sibirsk. Mat. Zh. 22 (1981), no. 4, 142–163, 231 (Russian). MR 624412
S. A. Nazarov, The Vishik-Lyusternik method for elliptic boundary value problems in regions with conic points. II. Problem in a bounded domain, Sibirsk. Mat. Zh. 22 (1981), no. 5, 132–152, 223 (Russian). MR 632823
N. Kh. Arutiunian and S. A. Nazarov, On singularities of the stress function at the corner points of the transverse cross section of a twisted bar with a thin reinforcing layer, Prikl. Mat. Mekh. 47 (1983), no. 1, 122–132 (Russian); English transl., J. Appl. Math. Mech. 47 (1983), no. 1, 94–103 (1984). MR 733746, DOI 10.1016/0021-8928(83)90040-0
Gabriel Caloz, Martin Costabel, Monique Dauge, and Grégory Vial, Asymptotic expansion of the solution of an interface problem in a polygonal domain with thin layer, Asymptot. Anal. 50 (2006), no. 1-2, 121–173. MR 2286939
S. A. Nazarov, Localization near the corner point of the first eigenfunction of the Dirichlet problem in a domain with thin edging, Sibirsk. Mat. Zh. 52 (2011), no. 2, 350–370 (Russian, with Russian summary); English transl., Sib. Math. J. 52 (2011), no. 2, 274–290. MR 2841554, DOI 10.1134/S003744661102011X
S. G. Mihlin, Lineĭ nye uravneniya v chastnykh proizvodnykh, Izdat. “Vysš. Škola”, Moscow, 1977 (Russian). MR 510535
V. G. Maz′ya and S. A. Nazarov, Singularities of solutions of the Neumann problem at a conic point, Sibirsk. Mat. Zh. 30 (1989), no. 3, 52–63, 218 (Russian); English transl., Siberian Math. J. 30 (1989), no. 3, 387–396 (1990). MR 1010835, DOI 10.1007/BF00971492
F. Ursell, Trapping modes in the theory of surface waves, Proc. Cambridge Philos. Soc. 47 (1951), 347–358. MR 41604, DOI 10.1017/s0305004100026700
—, Mathematical aspects of trapping modes in the theory of surface waves, J. Fluid Mech. 18 (1988), 495–503.
S. A. Nazarov, Sufficient conditions for the existence of trapped modes in problems of the linear theory of surface waves, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 369 (2009), no. Matematicheskie Voprosy Teorii Rasprostraneniya Voln. 38, 202–223, 227 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 167 (2010), no. 5, 713–725. MR 2749207, DOI 10.1007/s10958-010-9956-3
Giuseppe Cardone, Tiziana Durante, and Sergey A. Nazarov, Water-waves modes trapped in a canal by a near-surface rough body, ZAMM Z. Angew. Math. Mech. 90 (2010), no. 12, 983–1004. MR 2777549, DOI 10.1002/zamm.201000042
S. A. Nazarov, Localized surface waves in a periodic layer of a heavy fluid, Prikl. Mat. Mekh. 75 (2011), no. 2, 338–351 (Russian, with Russian summary); English transl., J. Appl. Math. Mech. 75 (2011), no. 2, 235–244. MR 2858614, DOI 10.1016/j.jappmathmech.2011.05.013
Serguei A. Nazarov and Jan Sokolowski, Shape sensitivity analysis of eigenvalues revisited, Control Cybernet. 37 (2008), no. 4, 999–1012. MR 2536484
Antoine Laurain, Sergey Nazarov, and Jan Sokolowski, Singular perturbations of curved boundaries in three dimensions. The spectrum of the Neumann Laplacian, Z. Anal. Anwend. 30 (2011), no. 2, 145–180. MR 2792999, DOI 10.4171/ZAA/1429