We study the asymptotic behaviour of eigenvalues of 2D vibrating systems with mass density perturbed in a vicinity of closed curves. The threshold case in which the resonance frequencies of the membrane and the frequencies of thin inclusion coincide is investigated. The perturbed eigenvalue problem can be realized as a family of self-adjoint operators acting on varying Hilbert spaces. However the so-called limit operator is non-self-adjoint and possesses the Jordan chains of length 2. Apart from the lack of self-adjointness, the operator has non-compact resolvent. As a consequence, its spectrum has a complicated structure, for instance, the spectrum contains a countable set of eigenvalues with infinite multiplicity. The complete asymptotic analysis of eigenvalues has been carried out.
W.Arendt and R.Mazzeo, Spectral properties of the Dirichlet-to-Neumann operator on Lipschitz domains, Ulmer Seminare Heft12 (2007), 23–38.
2.
F.S.Bates and G.H.Fredrickson, Block copolymer thermodynamics: Theory and experiment, Annual Review of Physical Chemistry41(1) (1990), 525–557. doi:10.1146/annurev.pc.41.100190.002521.
3.
C.Castro and E.Zuazua, Low frequency asymptotic analysis of a string with rapidly oscillating density, SIAM Journal on Applied Mathematics60(4) (2000), 1205–1233. doi:10.1137/S0036139997330635.
4.
C.Castro and E.Zuazua, High frequency asymptotic analysis of a string with rapidly oscillating density, European Journal of Applied Mathematics11(6) (2000), 595–622. doi:10.1017/S0956792500004307.
5.
G.A.Chechkin, Estimation of solutions of boundary-value problems in domains with concentrated masses located periodically along the boundary: Case of light masses, Mathematical Notes76(5–6) (2004), 865–879. doi:10.1023/B:MATN.0000049687.89273.d9.
6.
G.A.Chechkin, Asymptotic expansions of eigenvalues and eigenfunctions of an elliptic operator in a domain with many “light” concentrated masses situated on the boundary. Two-dimensional case, Izvestiya: Mathematics69(4) (2005), 805. doi:10.1070/IM2005v069n04ABEH001665.
7.
G.A.Chechkin and T.A.Mel’nyk, Asymptotics of eigenelements to spectral problem in thick cascade junction with concentrated masses, Applicable Analysis91(6) (2012), 1055–1095. doi:10.1080/00036811.2011.602634.
8.
G.A.Chechkin, M.E.Pérez and E.I.Yablokova, Non-periodic boundary homogenization and “light” concentrated masses, Indiana University Mathematics Journal (2005), 321–348. doi:10.1512/iumj.2005.54.2487.
9.
M.V.Fedoruyk, V.M.Babich, V.F.Lazutkin, … and B.R.Vainberg, Partial Differential Equations V: Asymptotic Methods for Partial Differential Equations (Vol. 5), Springer Science & Business Media, 1999.
10.
F.R.Gantmakher and M.G.Krein, Oscillation Matrices and Kernels and Small Oscillations of Mechanical Systems, Gostekhizdat, Moscow, 1950, English transl, American Mathematical Soc., 2002.
11.
Y.Golovaty, On spectrum of strings with -like perturbations of mass density, Visnyk Lviv. Univ., Ser. Mech. Math.89 (2020), 60–79.
12.
Y.Golovaty, D.Gómez, M.Lobo and E.Pérez, Asymptotics for the eigenelements of vibrating membranes with very heavy thin inclusions, C. R., Méc., Acad. Sci. Paris330(11) (2002), 777–782.
13.
Y.Golovaty and H.Hrabchak, Asymptotics of the spectrum of a Sturm–Liouville operator on networks with perturbed density, Visnyk Lviv. Univ., Ser. Mech. Math.67 (2007), 66–83.
14.
Y.Golovaty and H.Hrabchak, On Sturm–Liouville problem on starlike graphs with “heavy” nodes, Visnyk Lviv. Univ., Ser. Mech. Math.72 (2010), 63–78.
15.
Y.D.Golovaty, D.Gómez, M.Lobo and E.Pérez, On vibrating membranes with very heavy thin inclusions, Math. Models Methods Appl. Sci.14(7) (2004), 987–1034. doi:10.1142/S0218202504003520.
16.
Y.D.Golovaty and A.S.Lavrenyuk, Asymptotic expansions of local eigenvibrations for plate with density perturbed in neighbourhood of one-dimensional manifold, Matematychni Studii13(1) (2000), 51–62.
17.
Y.D.Golovatyi, Spectral properties of oscillatory systems with added masses, Trudy Moskov. Mat. Obshch.54(29–72) (1992), 278.
18.
Y.D.Golovatyi, S.A.Nazarov and O.A.Oleinik, Asymptotic expansions of eigenvalues and eigenfunctions in problems on oscillations of a medium with concentrated perturbations, Trudy Matematicheskogo Instituta imeni VA Steklova192 (1990), 42–60.
19.
Y.D.Golovatyi, S.A.Nazarov, O.A.Oleinik and T.S.Soboleva, Natural oscillations of a string with an additional mass, Sibirsk. Mat. Zh.29(5) (1988), 71–91, 237.
20.
D.Gómez, M.Lobo, S.A.Nazarov and E.Pérez, Asymptotics for the spectrum of the Wentzell problem with a small parameter and other related stiff problems, Journal de mathématiques pures et appliquées86(5) (2006), 369–402. doi:10.1016/j.matpur.2006.08.003.
21.
D.Gómez, M.Lobo, S.A.Nazarov and E.Pérez, Spectral stiff problems in domains surrounded by thin bands: Asymptotic and uniform estimates for eigenvalues, Journal de Mathématiques Pures et Appliquées85(4) (2006), 598–632. doi:10.1016/j.matpur.2005.10.013.
22.
D.Gómez, S.A.Nazarov and M.E.Pérez-Martínez, Localization effects for Dirichlet problems in domains surrounded by thin stiff and heavy bands, Journal of Differential Equations270 (2021), 1160–1195. doi:10.1016/j.jde.2020.09.011.
23.
T.Kato, Perturbation Theory for Linear Operators, 2nd edn, Grundlehren der Mathematischen Wissenschaften, Vol. 132, Springer-Verlag, Berlin-New York, 1976.
24.
B.D.Ladbrooke and D.Chapman, Thermal analysis of lipids, proteins and biological membranes a review and summary of some recent studies, Chemistry and Physics of Lipids3(4) (1969), 304–356. doi:10.1016/0009-3084(69)90040-1.
25.
M.Lobo and E.Perez, On vibrations of a body with many concentrated masses near the boundary, Math. Models Methods Appl. Sci.3 (1993), 249–273. doi:10.1142/S021820259300014X.
26.
M.Lobo and E.Perez, Vibrations of a body with many concentrated masses near the boundary: High frequency vibrations, in: Spectral Analysis of Complex Structures, Hermann, 1995, pp. 85–101.
27.
M.Lobo and E.Perez, Vibrations of a membrane with many concentrated masses near the boundary, Mathematical Models and Methods in Applied Sciences5(5) (1995), 565–585. doi:10.1142/S0218202595000334.
28.
M.Lobo and E.Pérez, Local problems for vibrating systems with concentrated masses: A review, Comptes Rendus Mécanique331(4) (2003), 303–317. doi:10.1016/S1631-0721(03)00058-5.
29.
T.A.Mel’nik, Vibrations of a thick periodic junction with concentrated masses, Mathematical Models and Methods in Applied Sciences11(06) (2001), 1001–1027. doi:10.1142/S0218202501001215.
30.
T.A.Mel’nik and S.A.Nazarov, Asymptotic analysis of the Neumann problem on the junction of a body and thin heavy rods, Algebra i Analiz12(2) (2000), 188–238, St. Petersburg Math. J., 12, 2, 2001, 317–351.
31.
S.A.Nazarov and E.Perez, New asymptotic effects for the spectrum of problems on concentrated masses near the boundary, Comptes Rendus Mécanique337(8) (2009), 585–590. doi:10.1016/j.crme.2009.07.002.
32.
S.A.Nazarov and M.E.Pérez, On multi-scale asymptotic structure of eigenfunctions in a boundary value problem with concentrated masses near the boundary, Revista Matemática Complutense31(1) (2018), 1–62. doi:10.1007/s13163-017-0243-4.
33.
O.A.Oleinik, A.S.Shamaev and G.A.Yosifian, Mathematical Problems in Elasticity and Homogenization, North-Holland, Amsterdam, 1992.
34.
R.Pratibha, W.Park and I.I.Smalyukh, Colloidal gold nanosphere dispersions in smectic liquid crystals and thin nanoparticle-decorated smectic films, Journal of Applied Physics107(6) (2010), 063511. doi:10.1063/1.3330678.
35.
M.D.Riva and L.Provenzano, On vibrating thin membranes with mass concentrated near the boundary: An asymptotic analysis, SIAM Journal on Mathematical Analysis50(3) (2018), 2928–2967. doi:10.1137/17M1118221.
36.
V.Rybalko, Vibrations of elastic systems with a large number of tiny heavy inclusions, Asymptotic Analysis32(1) (2002), 27–62.
37.
J.Sanchez Hubert and E.Sánchez-Palencia, Vibration and Coupling of Continuous Systems, Springer-Verlag, Berlin, 1989.
38.
E.Sánchez-Palencia, Perturbation of Eigenvalues in Thermoelasticity and Vibration of Systems with Concentrated Masses, In Trends and Applications of Pure Mathematics to Mechanics, Vol. 346, Springer, Berlin, Heidelberg, 1984.
39.
T.Sarpkaya, Wave Forces on Offshore Structures, Cambridge University Press, 2010.
40.
P.Sens and M.S.Turner, Inclusions in thin smectic films, Journal de Physique II7(12) (1997), 1855–1870. doi:10.1051/jp2:1997218.
41.
E.I.Yablokova, On vibrations of bodies with concentrated masses near boundaries, Journal of Mathematical Sciences129(1) (2005), 3688–3695. doi:10.1007/s10958-005-0308-7.
