Asymptotics of the spectrum of a boundary-value problem with the Steklov condition on small sets periodically distributed along a contour

S. A. Nazarov

Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 51, Tome 536 (2024), pp. 178-227

Résumé

We construct asymptotics of eigenvalues of the Laplace equation in a multi-dimesional domain with the spectral Steklov conditions on small identical sets distributed frequently and periodically along a smooth closed contour at a planar part of the boundary while its remaining part is supplied with the Dirichlet condition. We describe the localization effect for the eigenfunctions near the contour. The limiting spectral problem implies an ordinary differential equation at the contour whose coefficients depend quadratically on its curature. Asymptotic structures in the three-dimensional domain differ from other dimensions because they become dependent on logarithm of the small parameter, namely the period of distribution of the Steklov “spots”.

Bibliographie

[1] O. A. Ladyzhenskaya, N. N. Uraltseva, Lineinye i kvazilineinye uravneniya ellipticheskogo tipa, Nauka, M., 1973 | MR

[2] E. Sanchez-Palencia, “Boundary value problems in domains containing perforated walls”, Nonlinear Partial Differential Equations and Their Applications (Collège de France Seminar), v. III, Res. Notes in Math., 70, Pitman, Boston, 1982, 309–325 | MR

[3] A. Damlamian, Ta-Tsien Li, “Boundary homogenization for elliptic problems”, J. Math. Pures Appl., 66:4 (1987), 351–361 | MR | Zbl

[4] E. Pérez, “On periodic Steklov type eigenvalue problems on half-bands and the spectral homogenization problem”, Discrete Contin. Dyn. Syst. Ser. B, 7:4 (2007), 859–883 | MR

[5] A. G. Chechkina, “Usrednenie spektralnykh zadach s singulyarnym vozmuscheniem usloviya Steklova”, Izvestiya RAN. Ser. matem., 81:1 (2017), 203–240 | DOI | MR | Zbl

[6] D. Gomez, S. A. Nazarov , E. Perez, “Homogenization of Winkler–Steklov spectral conditions in three-dimensional linear elasticity”, Z. Angew. Math. Phys., 69:2 (2018), 35 | DOI | MR | Zbl

[7] R. R. Gadylshin, A. L. Pyatnitskii, G. A. Chechkin, “Ob asimptotikakh sobstvennykh znachenii kraevoi zadachi v ploskoi oblasti tipa sita Steklova”, Izvestiya RAN. Ser. matem., 82:6 (2018), 37–64 | DOI | MR | Zbl

[8] N. G. Kuznetsov, V. G. Mazia, B. R. Vainberg, Linear Water Waves: A mathematical approach, Cambridge Univ. Pr., Cambridge, 2002 | MR | Zbl

[9] V. Chiado Piat, S.A. Nazarov, “Steklov spectral problems in a set with a thin toroidal hole”, Partial Differential Equations in Applied Mathematics, 1 (2020), 100007 | DOI

[10] A. Herrot, Extremum problems for eigenvalues of elliptic operators, Birkhäuser Verlag, Basel, 2006 | MR

[11] V. A. Kondratev, “O gladkosti reshenii zadachi Dirikhle dlya uravneniya vtorogo poryadka v okrestnosti rebra”, Differentsialnye uravneniya, 6:10 (1970), 1831–1843 | Zbl

[12] V. G. Mazya, B. A. Plamenevskii, “Ob ellipticheskikh kraevykh zadachakh v oblastyakh s kusochno gladkoi granitsei”, Trudy simp. po mekh. sploshn. sred i rodstvennym probl. analiza, v. 1, Metsniereba, Tbilisi, 1973, 171–181

[13] V. A. Kondratev, “Osobennosti reshenii zadachi Dirikhle dlya uravneniya vtorogo poryadka v okrestnosti rebra”, Differentsialnye uravneniya, 13:11 (1977), 2026–2032 | MR | Zbl

[14] V. G. Maz'ja, J. Rosmann, “Über die Asymptotik der Lösungen elliptisher Randwertaufgaben in der Umgebung von Kanten”, Math. Nachr., 138 (1988), 27–53 | DOI | MR | Zbl

[15] S. A. Nazarov, B. A. Plamenevsky, Elliptic problems in domains with piecewise smooth boundaries, Walter de Gruyter, Berlin–New York, 1994 | MR

[16] S. A. Nazarov, “Vyvod variatsionnogo neravenstva dlya formy malogo prirascheniya treschiny otryva”, Mekhanika tverdogo tela, 1989, no. 2, 152–160

[17] E. Sanches-Palensiya, Neodnorodnye sredy i teoriya kolebanii, Mir, M., 1984 | MR

[18] N. S. Bakhvalov, G. P. Panasenko, Osrednenie prtsessov v periodicheskikh sredakh, Nauka, M., 1984

[19] V. V. Zhikov, S. M. Kozlov, O. A. Oleinik, Usrednenie differentsialnykh operatorov, Fizmatlit, M., 1993

[20] A. L. Pyatnitskii, G. A. Chechkin, A. S. Shamaev, Usrednenie. Metody i prilozheniya, Tamara Rozhkovskaya, Novosibirsk, 2007 | MR

[21] V. A. Kondratev, “Kraevye zadachi dlya ellipticheskikh uravnenii v oblastyakh s konicheskimi ili uglovymi tochkami”, Trudy Moskovsk. matem. obschestva, 16, 1963, 219–292

[22] V. A. Kozlov, V. G. Maz'ya, J. Rossman, Elliptic boundary value problems in domains with point singularities, Amer. Math. Soc., Providence, 1997 | MR | Zbl

[23] V. G. Mazya, B. A. Plamenevskii, “Otsenki v $L_p$ i v klassakh Geldera i printsip maksimuma Miranda-Agmona dlya reshenii ellipticheskikh kraevykh zadach v oblastyakh s osobymi tochkami na granitse”, Math. Nachr., 77 (1977), 25–82

[24] M. Vanninathan, “Homogenization of eigenvalue problems in perforated domains”, Proc. Indian Acad. Sci. Math. Sci., 90:3 (1981), 239–271 | DOI | MR | Zbl

[25] W. Kirsch, B. Simon, “Comparison theorems for the gap of Schrodinger operators”, J. Funct. Anal., 75:2 (1987), 396–410 | DOI | MR | Zbl

[26] T. A. Mel'nyk, “Vibrations of a thick periodic junction with concentrated masses”, Math. Models Methods Appl. Sci., 11:6 (2001), 1001–1027 | DOI | MR | Zbl

[27] S. A. Nazarov, M. E. Perez, “On multi-scale asymptotic structure of eigenfunctions in a boundary value problem with concentrated masses near the boundary”, Revista Matemática Complutense, 31:1 (2018), 1–62 | DOI | MR | Zbl

[28] E. B. Daners, B. Simon, “Ultracontractivity and the heat kernel for Schrodinger operators and Dirichlet Laplacians”, J. Funct. Anal., 59 (1984), 335–395 | DOI | MR

[29] M. Sh. Birman, T. A. Suslina, “Periodicheskie differentsialnye operatory vtorogo poryadka. Porogovye svoistva i usredneniya”, Algebra i Analiz, 15:5 (2003), 1–108

[30] S. A. Nazarov, “Polinomialnoe svoistvo samosopryazhennykh ellipticheskikh kraevykh zadach i algebraicheskoe opisanie ikh atributov”, Uspekhi matem. nauk, 54:5 (1999), 77–142 | DOI | MR | Zbl

[31] M. Sh. Birman, M. Z. Solomyak, Spektralnaya teoriya samosopryazhennykh operatorov v gilbertovom prostranstve, izd-vo Leningr. un-ta, L., 1980

[32] M. I. Vishik, L. A. Lyusternik, “Regulyarnoe vyrozhdenie i pogranichnyi sloi dlya lineinykh differentsialnykh uravnenii s malym parametrom”, Uspekhi matem. nauk, 12:5 (1957), 3–122 | Zbl

[33] L. Khermander, Lineinye differentsialnye operatory s chastnymi proizvodnymi, izd-vo Mir, M., 1965

[34] Maz'ya V., Nazarov S., Plamenevskij B., Asymptotic theory of elliptic boundary value problems in singularly perturbed domains, v. 1, 2, Birkhäuser Verlag, Basel, 2000 | MR | MR

[35] S. A. Nazarov, “Asimptotika sobstvennykh chisel zadachi Neimana pri kontsentratsii mass na tonkom toroidalnom mnozhestve”, Vestnik SPbGU Ser. 1, Vyp. 3, 2006, no. 15, 61–71

[36] I. M. Gelfand, “Razlozhenie po sobstvennym funktsiyam uravneniya s periodicheskimi koeffitsientami”, Doklady AN SSSR, 73 (1950), 1117–1120 | Zbl

[37] M. Reed, B. Simon, Methods of modern mathematical physics, v. 3, Academic Press Inc., New York, 1980 | MR | Zbl

[38] P. Kuchment, Floquet theory for partial differential equations, Birchäuser, Basel, 1993 | MR | Zbl

[39] I. V. Kamotskii, S. A. Nazarov, “O sobstvennykh funktsiyakh, lokalizovannykh okolo kromki tonkoi oblasti”, Problemy matem. analiza, 19, Nauchn. kniga, Novosibirsk, 1999, 105–148

[40] L. Friedlander, M. Solomyak, “On the spectrum of the Dirichlet Laplacian in a narrow strip”, Israel J. Math., 170 (2009), 337–354 | DOI | MR | Zbl

[41] D. Borisov, P. Freitas, “Singular asymptotic expansions for Dirichlet eigenvalues and eigenfunctions on thin planar domains”, Ann. Inst. Henri Poincaré. Anal. Non Linèaire, 26:2 (2009), 547–560 | MR | Zbl

[42] D. Borisov, P. Freitas, “Asymptotics of Dirichlet eigenvalues and eigenfunctions of the Laplacian on thin domains in $\mathbb{R}^d$”, J. Funct. Anal., 258:3 (2010), 893–912 | DOI | MR | Zbl

[43] S. A. Nazarov, E. Perez, J. Taskinen, “Localization effect for Dirichlet eigenfunctions in thin non-smooth domains”, Transactions of the American Mathematical Society, 368:7 (2016), 4787–4829 | DOI | MR | Zbl

[44] V. G. Mazya, S. A. Nazarov, A. B. Plamenevskii, “Ob asimptotike reshenii zadachi Dirikhle v trekhmernoi oblasti s vyrezannym tonkim telom”, Doklady AN SSSR, 256:1 (1981), 37–39 | MR | Zbl

[45] M. V. Fedoryuk, “Difraktsiya ploskoi volny na vytyanutom tele vrascheniya”, Doklady AN SSSR, 272:3 (1983), 587–590 | MR