On spectral properties of one boundary value problem with a surface energy dissipation

O. A. Andronova   ; V. I. Voytitskiy

Ufa mathematical journal, Tome 9 (2017) no. 2, pp. 3-16

Résumé

We study a spectral problem in a bounded domain ${\Omega \subset \mathbb{R}^{m}}$, depending on a bounded operator coefficient $Q>0$ and a dissipation parameter $\alpha>0$. In the general case we establish sufficient conditions ensuring that the problem has a discrete spectrum consisting of countably many isolated eigenvalues of finite multiplicity accumulating at infinity. We also establish the conditions, under which the system of root elements contains an Abel-Lidskii basis in the space $ L_2(\Omega)$. In model one- and two-dimensional problems we establish the localization of the eigenvalues and find critical values of $\alpha$.

Détail

Keywords: spectral parameter, quadratic operator pencil, localization of eigenvalues, compact operator, Schatten-von-Neumann classes $S_p$, Abel-Lidskii basis property.

Bibliographie

[1] Chueshov I.D., Vvedenie v teoriyu beskonechnomernykh dissipativnykh sistem, Akta, Kharkov, 2006, 433 pp.

[2] I. Chueshov, M. Eller, I. Lasiecka, “Finite Dimensionality of the Attractor for a Semilinear Wave Equation with Nonlinear Boundary Dissipation”, Comm. Partial Differential Equations, 29:11–12 (2004), 1847–1876 | MR | Zbl

[3] I. Chueshov, I. Lasiecka, “Global Attractors for von Karman Evolutions with a Nonliner Boundary Dissipations”, J. Differential Equations, 198 (2004), 196–231 | DOI | MR | Zbl

[4] J. Lagnese, “Decay of the Solution of the Wave Equation in a Bounded Region with Boundary Dissipation”, J. Diff. Equations, 50 (1983), 163–182 | DOI | MR | Zbl

[5] I. Lasiecka, D. Tataru, “Uniform Boundary Stabilization of Semilinear Wave Equation with Nonlinear Boundary Dissipation”, Diff. Integral Equations, 6 (1993), 507–533 | MR | Zbl

[6] A.A. Shkalikov, “Spectral analysis of the Regge problem”, Russ. J. Math. Phys., 8:3 (2001), 356–364 | MR | Zbl

[7] Shkalikov A.A., Shkred A.V., “Zadacha ob ustanovivshikhsya kolebaniyakh transversalno-izotropnogo polutsilindra”, Matem. sbornik, 182:8 (1991), 1222–1246 | Zbl

[8] Krein M.G., Langer G.K., “O nekotorykh matematicheskikh printsipakh teorii dempfirovannykh kolebanii kontinuumov”, Trudy Mezhdunar. simpoziuma po primeneniyu TFKP v mekhanike sploshnoi sredy, Nauka, M., 1965, 283–322 | Zbl

[9] Gokhberg I.Ts., Krein M.G., Vvedenie v teoriyu lineinykh nesamosopryazhennykh operatorov v gilbertovom prostranstve, Nauka, M., 1965, 448 pp.

[10] Andronova O.A., Kopachevskii N.D., “O lineinykh zadachakh s poverkhnostnoi dissipatsiei energii”, Sovremennaya matematika. Fundamentalnye napravleniya, 29, 2008, 11–28

[11] Russakovskii E.M., “Operatornaya traktovka granichnoi zadachi so spektralnym parametrom, polinomialno vkhodyaschim v granichnye usloviya”, Funktsioalnyi analiz i ego prilozheniya, 9:4 (1975), 91–92

[12] Shkalikov A.A., “Kraevye zadachi dlya obyknovennykh differentsialnykh uravnenii s parametrom v granichnykh usloviyakh”, Trudy seminara im. I.G. Petrovskogo, 9, 1983, 140–166

[13] J. Ercolano, M. Schechter, “Spectral Theory for Operators Generated by Elliptic Boundary Problems with Eigenvalue Parameter in Boundary Conditions I-II”, Comm. Pure and Appl. Math., 18 (1965), 83–105 | DOI | MR | Zbl

[14] Barkovskii V.V., “Razlozhenie po sobstvennym funktsiyam samosopryazhennykh operatorov, sootvetstvuyuschikh obschim ellipticheskim zadacham s sobstvennym znacheniem v granichnykh usloviyakh”, UMZh, 19:1 (1967), 9–24 | Zbl

[15] Komarenko O.N., “Rozvinennya za vlasnimi vektorami samospryazhenikh operatoriv, porodzhenikh zagalnoyu zadacheyu transmisiï”, Zbirnik prats Institutu matematiki NAN Ukraïni, 2:1 (2005), 127–157 (na ukr. yaz.)

[16] Kozhevnikov A.N., “Ob asimptotike sobstvennykh znachenii i polnote kornevykh vektorov operatora, porozhdennogo kraevoi zadachei s parametrom v kraevom uslovii”, DAN SSSR, 200:6 (1971), 1273–1276 | Zbl

[17] Kozhevnikov A.N., “Spektralnye zadachi dlya psevdodifferentsialnykh sistem ellipticheskikh po Duglisu-Nirenbergu i ikh prilozheniya”, Matem. Sbornik, 92(134):1(9) (1973), 60–88 | Zbl

[18] Kopachevskii N. D., Krein S. G., Ngo Zui Kan, Operatornye metody v lineinoi gidrodinamike. Evolyutsionnye i spektralnye zadachi, Nauka, M., 1989, 416 pp.

[19] Kopachevskii N.D., “Ob abstraktnoi formule Grina dlya troiki gilbertovykh prostranstv i ee prilozheniyakh k zadache Stoksa”, Tavricheskii vestnik informatiki i matematiki (TVIM), 2004, no. 2, 52–80

[20] Kopachevskii N.D., “Abstraktnaya formula Grina dlya smeshannykh kraevykh zadach”, Uchenye zapiski Tavricheskogo natsionalnogo universiteta im. V.I. Vernadskogo. Seriya “Matematika. Mekhanika. Informatika i Kibernetika”, 20(59):2 (2007), 3–12

[21] Voititskii V.I., Kopachevskii N.D., Starkov P.A., “Vspomogatelnye abstraktnye kraevye zadachi i zadachi sopryazheniya”, Sovremennaya matematika. Fundamentalnye napravleniya, 34, 2009, 5–44

[22] Kopachevskii N.D., Radomirskaya K.A., “Abstraktnye smeshannye kraevye i spektralnye zadachi sopryazheniya”, Uchenye zapiski Tavricheskogo natsionalnogo universiteta im. V.I. Vernadskogo. Seriya «Matematika. Mekhanika. Informatika i Kibernetika», 27(66):1 (2014), 58–64

[23] Kopachevskii N.D., “Ob abstraktnoi formule Grina dlya troiki gilbertovykh prostranstv i polutoralineinykh form”, Sovremennaya matematika. Fundamentalnye napravleniya, 57, 2015, 71–107

[24] Voititskii V.I., “O spektralnykh zadachakh, porozhdennykh linearizovannoi zadachei Stefana s usloviem Gibbsa-Tompsona”, Nelineinye granichnye zadachi, 17 (2007), 31–49 | Zbl

[25] V. Voytitsky, “On Some Class of Self-adjoint Boundary Value Problems with the Spectral Parameter in the Equations and the Boundary Conditions”, Spectral Theory, Mathematical System Theory, Evolution Equations, Differential and Difference Equations, 21st International Workshop on Operator Theory and Applications (Berlin, July 2010), Operator Theory: Advances and Applications, 221, Springer, Basel AG, 2012, 635–651 | MR

[26] Agranovich M.S., “Spektralnye zadachi dlya silno ellipticheskikh sistem vtorogo poryadka v oblastyakh s gladkoi i negladkoi granitsei”, Uspekhi matematicheskikh nauk, 57:5(347) (2002), 3–78 | DOI | Zbl

[27] E. Gagliardo, “Caratterizzazioni Delle Trace Sulla Frontiera Relative ad Alaine Classi di Funzioni in $n$ Variabili”, Rendiconti del Seminare Matematico della Universita di Padova, 27 (1957), 284–305 (In Italian) | MR | Zbl

[28] Birman M.Sh., Solomyak M.Z., Spektralnaya teoriya samosopryazhennykh operatorov v gilbertovom prostranstve, Lan, Sankt-Peterburg, 2010, 464 pp.

[29] Kostyuchenko A.G., Orazov M.B., “O nekotorykh svoistvakh kornei samosopryazhennogo kvadratichnogo puchka”, Funkts. analiz i ego prilozheniya, 9:4 (1975), 28–40

[30] Kostyuchenko A.G., Shkalikov A.A., “Samosopryazhennye kvadratichnye puchki operatorov i ellipticheskie zadachi”, Funkts. analiz i ego prilozheniya, 17:2 (1983), 38–61 | Zbl

[31] Keldysh M.V., Lidskii V.B., “Voprosy spektralnoi teorii nesamosopryazhennykh operatorov”, Trudy IV Vsesoyuznogo matem. s'ezda, v. 1, 1963, 101–120

[32] Lidskii V.B., “O summiruemosti ryadov po glavnym vektoram nesamosopryazhennykh operatorov”, Trudy Mosk. matem. ob-va, 1962, 3–35

[33] Markus A.S., “Nekotorye priznaki polnoty sistemy kornevykh vektorov lineinogo operatora v banakhovom prostranstve”, Matem. sbornik, 70 (112):4 (1966), 526–561

[34] M.S. Agranovich, B.Z. Katsenelenbaum, A.N. Sivov, N.N. Voitovich, Generalized Method of Egenoscillations in Difraction Theory, Viley-VCH, Berlin, 1999, 380 pp. | MR

[35] Agranovich M.S., “Spektralnye zadachi v lipshitsevykh oblastyakh”, SMFN, 39, 2011, 11–35

[36] Markushevich A.I., Teoriya analiticheskikh funktsii, v. 2, Nauka, M., 1968, 624 pp.