Geodesic
Operator-Norm Resolvent Asymptotic Analysis of Continuous Media with High-Contrast Inclusions
Matematičeskie zametki, Tome 111 (2022) no. 3, pp. 375-392.

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Using a generalization of the classical notion of Weyl $m$-function and related formulas for the resolvents of boundary-value problems, we analyze the asymptotic behavior of solutions to a “transmission problem” for a high-contrast inclusion in a continuous medium, for which we prove the operator-norm resolvent convergence to a limit problem of “electrostatic” type. In particular, our results imply the convergence of the spectra of high-contrast problems to the spectrum of the limit operator, with order-sharp convergence estimates. The approach developed in the paper is of a general nature and can thus be successfully applied in the study of other problems of the same type.
Keywords: extensions of symmetric operators, generalized boundary triples, boundary value problems, transmission problems. 
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A. V. Kiselev; L. O. Silva; K. D. Cherednichenko. Operator-Norm Resolvent Asymptotic Analysis of Continuous Media with High-Contrast Inclusions. Matematičeskie zametki, Tome 111 (2022) no. 3, pp. 375-392. http://geodesic.mathdoc.fr/item/MZM_2022_111_3_a4/

[1] R. Hempel, K. Lienau, “Spectral properties of the periodic media in large coupling limit”, Comm. Partial Differential Equations, 25:7-8 (2000), 1445–1470 | DOI | MR | Zbl

[2] V. V. Zhikov, “Ob odnom rasshirenii i primenenii metoda dvukhmasshtabnoi skhodimosti”, Matem. sb., 191:7 (2000), 31–72 | DOI | MR | Zbl

[3] K. D. Cherednichenko, Yu. Ershova, A. V. Kiselev, “Effective behaviour of critical-contrast PDEs: micro-resonances frequency conversion, and time dispersive properties. I”, Comm. Math. Phys., 375:3 (2020), 1833–1884 | DOI | MR | Zbl

[4] V. V. Zhikov, “O lakunakh v spektre nekotorykh divergentnykh ellipticheskikh operatorov s periodicheskimi koeffitsientami”, Algebra i analiz, 16:5 (2004), 34–58 | MR | Zbl

[5] H. Ammari, H. Kang, K. Kim, H. Lee, “Strong convergence of the solutions of the linear elasticity and uniformity of asymptotic expansions in the presence of small inclusions”, J. Differential Equations, 254:12 (2013), 4446–4464 | DOI | MR | Zbl

[6] K. D. Cherednichenko, A. V. Kiselev, “Norm-resolvent convergence of one-dimensional high-contrast periodic problems to a Kronig-Penney dipole-type model”, Comm. Math. Phys., 349:2 (2017), 441–480 | DOI | MR | Zbl

[7] K. D. Cherednichenko, Yu. Yu. Ershova, A. V. Kiselev, “Time-dispersive behaviour as a feature of critical contrast media”, SIAM J. Appl. Math., 79:2 (2019), 690–715 | DOI | MR | Zbl

[8] K. Cherednichenko, Y. Ershova, A. Kiselev, S. Naboko, “Unified approach to critical-contrast homogenisation with explicit links to time-dispersive media”, Tr. MMO, 80, no. 2, MTsNMO, M., 2019, 295–342 | MR

[9] V. Ryzhov, “Spectral boundary value problems and their linear operators”, Analysis as a Tool in Mathematical Physics, Oper. Theory Adv. Appl., 276, Birkhäuser, Cham, 2020, 576–626 | MR | Zbl

[10] M. Sh. Birman, “K teorii samosopryazhennykh rasshirenii polozhitelno opredelennykh operatorov”, Matem. sb., 38 (80):4 (1956), 431–450 | MR | Zbl

[11] M. G. Krein, “Teoriya samosopryazhennykh rasshirenii poluogranichennykh ermitovykh operatorov i ee prilozheniya. I”, Matem. sb., 20 (62):3 (1947), 431–495 | MR | Zbl

[12] M. G. Krein, “Teoriya samosopryazhennykh rasshirenii poluogranichennykh ermitovykh operatorov i ee prilozheniya. II”, Matem. sb., 21 (63):3 (1947), 365–404 | MR | Zbl

[13] M. I. Vishik, “Ob obschikh kraevykh zadachakh dlya ellipticheskikh differentsialnykh uravnenii”, Tr. MMO, 1, GITTL, M.–L., 1952, 187–246 | MR | Zbl

[14] K. Cherednichenko, A. Kiselev, L. Silva, “Scattering theory for non-selfadjoint extensions of symmetric operators”, Analysis as a Tool in Mathematical Physics, Oper. Theory Adv. Appl., 276, Birkhäuser, Cham, 2020, 194–230 | MR | Zbl

[15] K. D. Cherednichenko, A. V. Kiselev, L. O. Silva, “Functional model for extensions of symmetric operators and applications to scattering theory”, Netw. Heterog. Media, 13:2 (2018), 191–215 | DOI | MR | Zbl

[16] K. D. Cherednichenko, A. V. Kiselev, L. O. Silva, “Functional model for boundary-value problems”, Mathematika, 67:3 (2021), 596–626 | DOI | MR

[17] B. S. Pavlov, “Teoriya dilatatsii i spektralnyi analiz nesamosopryazhennykh differentsialnykh operatorov”, Matematicheskoe programmirovanie i smezhnye voprosy. Teoriya operatorov v lineinykh prostranstvakh, TsEMI, M., 1976, 3–69

[18] S. N. Naboko, “Funktsionalnaya model teorii vozmuschenii i ee prilozheniya k teorii rasseyaniya”, Kraevye zadachi matematicheskoi fiziki. 10, Sbornik rabot, Tr. MIAN SSSR, 147, 1980, 86–114 | MR | Zbl

[19] G. P. Panasenko, “Asimptotika reshenii i sobstvennykh znachenii ellipticheskikh uravnenii s silno izmenyayuschimisya koeffitsientami”, Dokl. AN SSSR, 252:6 (1980), 1320–1325 | MR | Zbl

[20] V. I. Gorbachuk, M. L. Gorbachuk, Granichnye zadachi dlya differentsialno-operatornykh uravnenii, Naukova dumka, Kiev, 1984 | MR | Zbl

[21] M. Schechter, “A generalization of the problem of transmission”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3), 14 (1960), 207–236 | MR | Zbl

[22] M. Brown, M. Marletta, S. Naboko, I. Wood, “Boundary triples and $M$-functions for non-selfadjoint operators, with applications to elliptic PDEs and block operator matrices”, J. Lond. Math. Soc. (2), 77:3 (2008), 700–718 | DOI | MR | Zbl

[23] C. Tretter, Spectral Theory of Block Operator Matrices and Applications, Imperial College Press, London, 2000 | MR

[24] I. Schur, “Neue Begründung der Theorie der Gruppencharaktere”, Sitzungsberichte der Preussischen Akademie der Wissenschaften, Physikalisch-Mathematische Klasse, 1905, 406–432

[25] M. A. Naimark, “Spektralnye funktsii simmetricheskogo operatora”, Izv. AN SSSR. Ser. matem., 4:3 (1940), 277–318 | MR | Zbl

[26] M. A. Naimark, “Polozhitelno-opredelennye operatornye funktsii na kommutativnoi gruppe”, Izv. AN SSSR. Ser. matem., 7:5 (1943), 237–244 | MR | Zbl

[27] A. A. Shkalikov, “Kraevye zadachi dlya obyknovennykh differentsialnykh uravnenii s parametrom v granichnykh usloviyakh”, Tr. sem. im. I.G. Petrovskogo, 9 (1983), 140–179 | MR

[28] A. V. Shtraus, “Obobschennye rezolventy simmetricheskikh operatorov”, Izv. AN SSSR. Ser. matem., 18:1 (1954), 51–86 | MR | Zbl

[29] A. V. Shtraus, “Funktsionalnye modeli i obobschennye spektralnye funktsii simmetricheskikh operatorov”, Algebra i analiz, 10:5 (1998), 1–76 | MR | Zbl

[30] M. Sh. Birman M. Z. Solomyak, Spektralnaya teoriya samosopryazhennykh operatorov v gilbertovom prostranstve, Izd-vo LGU, L., 1980 | MR

[31] J. Albert, “Genericity of simple eigenvalues for elliptic PDE's”, Proc. Amer. Math. Soc., 48 (1975), 413–418 | MR | Zbl

[32] U. Smilansky, “Semiclassical quantization of chaotic billiards”, Chaos and Quantum Chaos, Lecture Notes in Phys., 411, Springer, Berlin, 1993, 57–120 | MR

[33] J. P. Eckmann, C. A. Pillet, “Spectral duality for planar billiards”, Comm. Math. Phys., 170:2 (1995), 283–313 | DOI | MR | Zbl

[34] A. B. Mikhailova, B. S. Pavlov, “Remark on the compensation of singularities in Krein's formula”, Methods of Spectral Analysis in Mathematical Physics, Oper. Theory Adv. Appl., 186, Birkhäuser Verlag, Basel, 2009, 325–337 | MR | Zbl