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Plummeting and blinking eigenvalues of the Robin Laplacian in a cuspidal domain

Published online by Cambridge University Press:  14 August 2019

Sergei A. Nazarov ,
Nicolas Popoff  and
Jari Taskinen
Sergei A. Nazarov
Affiliation:
Saint-Petersburg State University, Universitetskaya nab., 7–9, St. Petersburg, 199034Russia ([email protected]; [email protected])
Nicolas Popoff
Affiliation:
Institut de Mathématiques de Bordeaux UMR 5251, Université de Bordeaux, 351 cours dela Libération – F 33 405 Talence, France ([email protected])
Jari Taskinen
Affiliation:
Department of Mathematics and Statistics, P.O.Box 68, University of Helsinki, 00014 Helsinki, Finland ([email protected])
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Abstract

We consider the Robin Laplacian in the domains Ω and Ωε, ε > 0, with sharp and blunted cusps, respectively. Assuming that the Robin coefficient a is large enough, the spectrum of the problem in Ω is known to be residual and to cover the whole complex plane, but on the contrary, the spectrum in the Lipschitz domain Ωε is discrete. However, our results reveal the strange behaviour of the discrete spectrum as the blunting parameter ε tends to 0: we construct asymptotic forms of the eigenvalues and detect families of ‘hardly movable’ and ‘plummeting’ ones. The first type of the eigenvalues do not leave a small neighbourhood of a point for any small ε > 0 while the second ones move at a high rate O(| ln ε|) downwards along the real axis ℝ to −∞. At the same time, any point λ ∈ ℝ is a ‘blinking eigenvalue’, i.e., it belongs to the spectrum of the problem in Ωε almost periodically in the | ln ε|-scale. Besides standard spectral theory, we use the techniques of dimension reduction and self-adjoint extensions to obtain these results.

Keywords

Laplace operatorrobin conditionspectral problemcuspidal domaineigenvalueresidual spectrum
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 6 , December 2020 , pp. 2871 - 2893
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Copyright
Copyright © 2019 The Royal Society of Edinburgh

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References

1Arendt, W. and Warma, M.. The Laplacian with Robin boundary conditions on arbitrary domains. Potential Anal. 19 (2003), 341–363.CrossRefGoogle Scholar
2Birman, M. S. and Solomyak, M. Z.. Spectral theory of selfadjoint operators in Hilbert space (Leningrad: Leningrad Univ., 1980). (English transl. Math. Appl. (Soviet Ser.), D. Reidel Publishing Co., Dordrecht (1987)).Google Scholar
3Daners, D.. Robin boundary value problems on arbitrary domains. Trans. Amer. Math. Soc. 352 (2000), 4207–4236.CrossRefGoogle Scholar
4Daners, D.. Principal eigenvalues for generalised indefinite Robin problems. Potential Anal. 38 (2013), 1047–1069.Google Scholar
5de Snoo, H., Fleige, A., Hassi, S. and Winkler, H.. Non-semi-bounded closed symmetric forms associated with a generalized Friedrichs extension. Proc. Roy. Soc. Edinburgh Sect. A 144 (2014), 731–745.CrossRefGoogle Scholar
6Evans, D., Levitin, M. and Vasil'ev, D.. Existence theorems for trapped modes. J. Fluid Mech. 261 (1994), 21–31.CrossRefGoogle Scholar
7Gohberg, I. M. and Krein, M. G.. Introduction to the Theory of Linear Nonselfadjoint Operators (Providence, RI: Am. Math. Soc., 1969).Google Scholar
8Kamotskii, I. V. and Nazarov, S. A.. Spectral problems in singularly perturbed domains and self-adjoint extensions of differential operators. Trudy St.-Petersburg Mat. Obshch. 6 (1998), 151–212. (English transl. Translations Am. Math. Soc. Ser. 2 199 (2000), 127–181.)Google Scholar
9Kato, T.. Perturbation Theory of Linear Operators, 2nd corrected edn (Berlin: Springer-Verlag, 1995).CrossRefGoogle Scholar
10Kondratiev, V. A.. Boundary value problems for elliptic problems in domains with conical or corner points. Trudy Moskov. Matem. Obshch. 16 (1967), 209–292. (English transl. Trans. Moscow Math. Soc. 16 (1967), 227–313.)Google Scholar
11Kovarik, H. and Pankrashkin, K.. Robin eigenvalues on domains with peaks, arXiv preprint arXiv:1803.09295 (2018).Google Scholar
12Kozlov, V. and Maz'ya, V.. Differential Equations with Operator Coefficients with Applications to Boundary Value Problems for Partial Differential Equations. Springer Monographs in Mathematics (Berlin: Springer-Verlag, 1999).Google Scholar
13Kozlov, V. A., Maz'ya, V. G. and Rossmann, J.. Elliptic Boundary Value Problems in Domains with Point Singularities (Providence: Amer. Math. Soc., 1997).Google Scholar
14Ladyzhenskaya, O. A.. Boundary Value Problems of Mathematical Physics (New York: Springer-Verlag, 1985).CrossRefGoogle Scholar
15Maz'ja, V. G. and Plamenevskii, B. A.. Estimates in L p and Hölder classes and the Miranda-Agmon maximum principle for solutions of elliptic boundary value problems in domains with singular points on the boundary. Math. Nachr. 81 (1978), 25–82. (Engl. Transl. in: Amer. Math. Soc. Transl. (Ser. 2) 123 (1984), 1–56.)Google Scholar
16Maz'ya, V. G. and Poborchi, S. V.. Differentiable Functions on Bad Domains (River Edge, NJ: World Scientific Publishing, 1997).CrossRefGoogle Scholar
17Maz'ya, V. G., Nazarov, S. A. and Plamenevskii, B. A.. Asymptotische Theorie elliptischer Randwertaufgaben in singulär gestörten Gebieten. 1 (Berlin: Akademie-Verlag, 1991). (English transl.: Maz'ya V. G., Nazarov S. A., Plamenevskij B. A., Asymptotic theory of elliptic boundary value problems in singularly perturbed domains. Vol. 1. Birkhäuser Verlag, Basel (2000)).Google Scholar
18McIntosh, A. G. R.. Hermitian bilinear forms which are not semibounded. Bull. Am. Math. Soc. 76 (1970), 732–737.CrossRefGoogle Scholar
19Nazarov, S. A.. The polynomial property of self-adjoint elliptic boundary-value problems and the algebraic description of their attributes. Uspehi Mat. Nauk. 54 (1999), 77–142. (English transl.: Russ. Math. Surveys. 54, 5. (1999), 947-1014).Google Scholar
20Nazarov, S. A. and Plamenevsky, B. A.. Elliptic Problems in Domains with Piecewise Smooth Boundaries (Berlin, New York: Walter de Gruyter, 1994).CrossRefGoogle Scholar
21Nazarov, S. A. and Taskinen, J.. On the spectrum of the Steklov problem in a domain with a peak. Vestnik St. Petersburg Univ. Math. 41 (2008), 45–52.CrossRefGoogle Scholar
22Nazarov, S. A. and Taskinen, J.. On the spectrum of the Robin problem in a domain with a peak. Funkt. Anal. i Prilozhen. 45 (2011), 93–96. (English transl. Funct. Anal. Appl. 45, 1 (2011) 77–79.)Google Scholar
23Nazarov, S. A. and Taskinen, J.. Radiation conditions at the top of a rotational cusp in the theory of water-waves. Math. Model. Numer. Anal. 45 (2011), 947–979.Google Scholar
24Nazarov, S. A. and Taskinen, J.. Spectral anomalies of the Robin Laplacian in non-Lipschitz domains. J. Math. Sci. Univ. Tokyo. 20 (2013), 27–90.Google Scholar
25Nazarov, S. A. and Taskinen, J.. ‘Blinking’ eigenvalues of the Steklov problem generate the continuous spectrum in a cuspidal domain. Submitted.Google Scholar
26Rofe-Beketov, F. S.. Self-adjoint extensions of differential operators in the space of vector functions. Dokl. Akad. Nauk SSSR 184 (1969), 1034–1037 (English transl. Soviet Math. Dokl., 10 (1969), 188–192).Google Scholar
27Višik, M. I. and Ljusternik, L. A.. Regular degeneration and boundary layer for linear differential equations with small parameter. Am. Math. Soc. Transl. 20 (1962), 239–364.CrossRefGoogle Scholar