×

Multiple eigenvalues for the Steklov problem in a domain with a small hole. A functional analytic approach. (English) Zbl 1473.35373

Summary: Let \(\alpha\in]0,1[\). Let \(\Omega^o\) be a bounded open domain of \(\mathbb{R}^n\) of class \(C^{1,\alpha}\). Let \(\nu_{\Omega}^o\) denote the outward unit normal to \(\partial\Omega^o\). We assume that the Steklov problem \(\Delta u=0\) in \(\Omega^o\), \(\frac{\partial u}{\partial\nu_\Omega^o}=\lambda u\) on \(\partial\Omega^o\) has a multiple eigenvalue \(\widetilde{\lambda}\) of multiplicity \(r\). Then we consider an annular domain \(\Omega(\epsilon)\) obtained by removing from \(\Omega^o\) a small cavity of class \(C^{1,\alpha}\) and size \(\epsilon>0\), and we show that under appropriate assumptions each elementary symmetric function of \(r\) eigenvalues of the Steklov problem \(\Delta u=0\) in \(\Omega(\epsilon)\), \(\frac{\partial u}{\partial\nu_{\Omega(\epsilon)}}=\lambda u\) on \(\partial\Omega(\epsilon)\) which converge to \(\widetilde{\lambda}\) as \(\varepsilon\) tend to zero, equals real a analytic function defined in an open neighborhood of \((0,0)\) in \(\mathbb{R}^2\) and computed at the point \((\epsilon,\delta_{2,n}\epsilon\log\epsilon)\) for \(\epsilon>0\) small enough. Here \(\nu_{\Omega(\epsilon)}\) denotes the outward unit normal to \(\partial \Omega(\epsilon)\), and \(\delta_{2,2}\equiv 1\) and \(\delta_{2,n}\equiv 0\) if \(n\geqslant 3\). Such a result is an extension to multiple eigenvalues of a previous result obtained for simple eigenvalues in collaboration with S. Gryshchuk.

MSC:

35P15 Estimates of eigenvalues in context of PDEs
35B25 Singular perturbations in context of PDEs
35J25 Boundary value problems for second-order elliptic equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] D. Buoso and P.D. Lamberti, Shape sensitivity analysis of the eigenvalues of the Reissner-Mindlin system,SIAM J. Math. Anal.47(1) (2015), 407-426. doi:10.1137/140969968. · Zbl 1322.35100
[2] H. Cartan,Cours de Calcul Différentiel, Hermann, Paris, 1967. · Zbl 0156.36102
[3] V. Chiado Piat, S. Nazarov and A.L. Piatnitski, Steklov problems in perforated domains with a coefficient of indefinite sign,Networks and heterogeneous media7(2012), 151-178. doi:10.3934/nhm.2012.7.151. · Zbl 1262.35025
[4] M. Dalla Riva, M. Lanza de Cristoforis and P. Musolino,A functional analytic approach to singularly perturbed boundary value problems, 2020, typewritten manuscript. · Zbl 1357.35112
[5] K. Deimling,Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985. · Zbl 0559.47040
[6] B. Dittmar, Zu einem Stekloffschen Eigenwertproblem in Ringgebieten,Mitt. Math. Sem. Giessen228(1996), 1-7. · Zbl 0866.35082
[7] F. Dondi and M. Lanza de Cristoforis, Regularizing properties of the double layer potential of second order elliptic differential operators,Mem. Differ. Equ. Math. Phys.71(2017), 69-110. · Zbl 1392.31007
[8] H. Douanla, Two-scale convergence of Stekloff eigenvalue problems in perforated domains,Bound. Value Probl.2010 (2010), Article ID 853717. doi:10.1155/2010/853717. · Zbl 1214.47020
[9] N. Dunford and J.T. Schwartz,Linear Operators. I. General Theory. With the Assistance of W. G. Bade and R.G. Bartle, Interscience Publishers, Inc., New York; Interscience Publishers, Ltd., London, 1958. · Zbl 0084.10402
[10] G.B. Folland,Introduction to Partial Differential Equations, 2nd edn, Princeton University Press, Princeton N.J., 1995. · Zbl 0841.35001
[11] D. Gilbarg and N.S. Trudinger,Elliptic Partial Differential Equations of Second Order, Springer Verlag, Berlin, 1983. · Zbl 0562.35001
[12] S. Gryshchuk and M. Lanza de Cristoforis, Simple eigenvalues for the Steklov problem in a domain with a small hole. A functional analytic approach,Mathematical Methods in the Applied Sciences37(2014), 1755-1771. doi:10.1002/mma. 2933. · Zbl 1301.35065
[13] A. Henrot,Extremum Problems for Eigenvalues of Elliptic Operators, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2006. · Zbl 1109.35081
[14] E. Hille and R.S. Phillips,Functional Analysis and Semigroups, Amer. Math. Soc. Colloq. Publ., Vol. 31, 1957. · Zbl 0078.10004
[15] T. Kato,Perturbation Theory for Linear Operators, Springer Verlag, Berlin, New York, 1976. M. Lanza de Cristoforis / Multiple Steklov eigenvalues365 · Zbl 0836.47009
[16] N.G. Kuznetsov and O.V. Motygin, The Steklov problem in symmetric domains with infinitely extended boundary, in: Proceedings of the St. Petersburg Mathematical Society. XIV, Amer. Math. Soc. Transl. Ser. 2, Vol. 228, Amer. Math. Soc., Providence, RI, 2009, pp. 45-78. · Zbl 1191.35190
[17] P.D. Lamberti, Steklov-type eigenvalues associated with best Sobolev trace constants: Domain perturbation and overdetermined systems,Complex Var. Elliptic Equ.59(3) (2014), 309-323. doi:10.1080/17476933.2011.557155. · Zbl 1287.35048
[18] P.D. Lamberti and M. Lanza de Cristoforis, A real analyticity result for symmetric functions of the eigenvalues of a domain dependent Dirichlet problem for the Laplace operator,Journal of Nonlinear and Convex Analysis5(2004), 19-42. · Zbl 1059.35090
[19] P.D. Lamberti and M. Lanza de Cristoforis, A global Lipschitz continuity result for a domain dependent Dirichlet eigenvalue problem for the Laplace operator,Zeitschrift für Analysis und ihre Anwendungen24(2005), 277-304. doi:10.4171/ ZAA/1240. · Zbl 1094.35086
[20] M. Lanza de Cristoforis, Properties and pathologies of the composition and inversion operators in Schauder spaces,Rend. Accad. Naz. Sci. XL15(1991), 93-109. · Zbl 0829.47059
[21] M. Lanza de Cristoforis, Asymptotic behaviour of the solutions of a nonlinear Robin problem for the Laplace operator in a domain with a small hole: A functional analytic approach,Complex Var. Elliptic Equ.52(2007), 945-977. doi:10.1080/ 17476930701485630. · Zbl 1143.35057
[22] M. Lanza de Cristoforis and P. Musolino, A real analyticity result for a nonlinear integral operator,J. Integral Equations Appl.25(2013), 21-46. doi:10.1216/JIE-2013-25-1-21. · Zbl 1278.47057
[23] M. Lanza de Cristoforis and L. Rossi, Real analytic dependence of simple and double layer potentials upon perturbation of the support and of the density,J. Integral Equations Appl.16(2004), 137-174. doi:10.1216/jiea/1181075272. · Zbl 1094.31001
[24] T.A. Mel’nik, Asymptotic expansions of eigenvalues and eigenfunctions for elliptic boundary-value problems with rapidly oscillating coefficients in a perforated cube,J. Math. Sci.75(1995), 1646-1671. doi:10.1007/BF02368668. · Zbl 0844.35072
[25] C. Miranda, Sulle proprietà di regolarità di certe trasformazioni integrali,Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez I7(1965), 303-336. · Zbl 0183.12701
[26] C. Miranda,Partial Differential Equations of Elliptic Type · Zbl 0198.14101
[27] S.A. Nazarov, Asymptotic behavior of the Steklov spectral problem in a domain with a blunted peak,Mat. Zametki86 (2009), 571-587, (Russian), English transl. inMath. Notes86(2009), 542-555. doi:10.4213/mzm4157. · Zbl 1185.35156
[28] S.A. Nazarov, On the spectrum of the Steklov problem in peak-shaped domains, in:Proceedings of the St. Petersburg Mathematical Society. XIV, Amer. Math. Soc. Transl. Ser. 2, Vol. 228, Amer. Math. Soc., Providence, RI, 2009, pp. 79- 131. · Zbl 1183.35212
[29] S.A. Nazarov, Asymptotic expansions of the eigenvalues of the Steklov problem in singularly perturbed domains,Algebra i Analiz26(2014), 119-184, (Russian), English transl. inSt. Petersburg Math. J., J.26(2015), 273-318. · Zbl 1456.35149
[30] S.A. Nazarov and J. Taskinen, On the spectrum of the Steklov problem in a domain with a peak,Vestnik St. Petersburg Univ. Math.41(2008), 45-52. doi:10.3103/S1063454108010081. · Zbl 1171.35086
[31] S.E. Pastukhova, On the error of averaging for the Steklov problem in a punctured domain,Differentsial’nye Uravneniya 31(1995), 1042-1054, (Russian), translation inDifferential Equations31(1995), 975-986. · Zbl 0861.35069
[32] G. Prodi and A. Ambrosetti,Analisi Non Lineare, Editrice Tecnico Scientifica, Pisa, 1973. · Zbl 0352.47001
[33] F. Rellich,Perturbation Theory of Eigenvalue Problems, Gordon and Breach Science Publ., New York, 1969. · Zbl 0181.42002
[34] W. Rudin,Functional Analysis, 2nd edn, McGraw-Hill, Inc., New York, 1991. · Zbl 0867.46001
[35] S.E. Shamma, Asymptotic behavior of Stekloff eigenvalues and eigenfunctions,SIAM J. Appl. Math.20(1971), 482-490. doi:10.1137/0120050. · Zbl 0216.38402
[36] M. Vanninathan, Homogenization of eigenvalue problems in perforated domains,Proc. Indian Acad. Sci. Math. Sci.90 (1981), 239-271. doi:10.1007/BF02838079 · Zbl 0486.35063
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.