On the asymptotics of solutions to the Neumann problem for hyperbolic systems in domains with conical points
HTML articles powered by AMS MathViewer
- by
A. Kokotov and B. Plamenevskiĭ
Translated by: B. A. Plamenevskiĭ - St. Petersburg Math. J. 16 (2005), 477-506
- DOI: https://doi.org/10.1090/S1061-0022-05-00862-9
- Published electronically: May 2, 2005
- PDF | Request permission
Abstract:
Hyperbolic systems of second-order differential equations are considered in a domain with conical points at the boundary; in particular, the equations of elastodynamics are discussed. The asymptotics of solutions near conical points is studied. The “hyperbolic character” of the asymptotics shows itself in the properties of the coefficients (stress intensity factors) depending on time. Some formulas for the coefficients are presented and sharp estimates in Soboloev’s norms are proved.References
- J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Vol. 1, Travaux et Recherches Mathématiques, No. 17, Dunod, Paris, 1968 (French). MR 0247243
- V. A. Kondrat′ev and O. A. Oleĭnik, Boundary value problems for a system in elasticity theory in unbounded domains. Korn inequalities, Uspekhi Mat. Nauk 43 (1988), no. 5(263), 55–98, 239 (Russian); English transl., Russian Math. Surveys 43 (1988), no. 5, 65–119. MR 971465, DOI 10.1070/RM1988v043n05ABEH001945
- Handbuch der Physik. Band VIa/2: Festkörpermechanik. II, Springer-Verlag, Berlin-New York, 1972 (German). Herausgegeben von S. Flügge; Bandherausgeber: C. Truesdell. MR 0347187
- V. A. Kozlov and V. G. Maz′ya, Spectral properties of operator pencils generated by elliptic boundary value problems in a cone, Funktsional. Anal. i Prilozhen. 22 (1988), no. 2, 38–46, 96 (Russian); English transl., Funct. Anal. Appl. 22 (1988), no. 2, 114–121. MR 947604, DOI 10.1007/BF01077601
- B. A. Plamenevskiĭ, On the Dirichlet problem for the wave equation in a cylinder with edges, Algebra i Analiz 10 (1998), no. 2, 197–228 (Russian); English transl., St. Petersburg Math. J. 10 (1999), no. 2, 373–397. MR 1629407
- A. Yu. Kokotov and B. A. Plamenevskiĭ, On the Cauchy-Dirichlet problem for a hyperbolic system in a wedge, Algebra i Analiz 11 (1999), no. 3, 140–195 (Russian); English transl., St. Petersburg Math. J. 11 (2000), no. 3, 497–534. MR 1711368
- Sergey A. Nazarov and Boris A. Plamenevsky, Elliptic problems in domains with piecewise smooth boundaries, De Gruyter Expositions in Mathematics, vol. 13, Walter de Gruyter & Co., Berlin, 1994. MR 1283387, DOI 10.1515/9783110848915.525
- A. Yu. Kokotov, P. Neĭttaanmyaki, and B. A. Plamenevskiĭ, Problems of diffraction by a cone: asymptotic behavior of the solutions near the vertex, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 259 (1999), no. Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funkts. 30, 122–144, 297–298 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (New York) 109 (2002), no. 5, 1894–1910. MR 1754360, DOI 10.1023/A:1014492224655
- A. Yu. Kokotov, P. Neittaämaki, and B. A. Plamenevskii, The Neumann problem for the wave equation in a cone, J. Math. Sci. (New York) 102 (2000), no. 5, 4400–4428. Function theory and applications. MR 1807064, DOI 10.1007/BF02672898
- V. G. Maz′ja and B. A. Plamenevskiĭ, The coefficients in the asymptotic expansion of the solutions of elliptic boundary value problems in a cone, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 52 (1975), 110–127, 219 (Russian, with English summary). Boundary value problems of mathematical physics and related questions of the theory of functions, 8. MR 0407445
- V. A. Kozlov, V. G. Maz′ya, and J. Rossmann, Elliptic boundary value problems in domains with point singularities, Mathematical Surveys and Monographs, vol. 52, American Mathematical Society, Providence, RI, 1997. MR 1469972, DOI 10.1090/surv/052
- S. A. Nazarov and B. A. Plamenevskiĭ, The Neumann problem for selfadjoint elliptic systems in a domain with a piecewise-smooth boundary, Trudy Leningrad. Mat. Obshch. 1 (1990), 174–211, 246–247 (Russian). MR 1104210
- Gregory Eskin, The wave equation in a wedge with general boundary conditions, Comm. Partial Differential Equations 17 (1992), no. 1-2, 99–160. MR 1151258, DOI 10.1080/03605309208820836
- Jeff Cheeger and Michael Taylor, On the diffraction of waves by conical singularities. I, Comm. Pure Appl. Math. 35 (1982), no. 3, 275–331. MR 649347, DOI 10.1002/cpa.3160350302
- Motoo Uchida, Microlocal analysis of diffraction by a corner, Ann. Sci. École Norm. Sup. (4) 25 (1992), no. 1, 47–75. MR 1152613
- Patrick Gérard and Gilles Lebeau, Diffusion d’une onde par un coin, J. Amer. Math. Soc. 6 (1993), no. 2, 341–424 (French, with English summary). MR 1157289, DOI 10.1090/S0894-0347-1993-1157289-8
- P. Grisvard, Contrôlabilité exacte des solutions de l’équation des ondes en présence de singularités, J. Math. Pures Appl. (9) 68 (1989), no. 2, 215–259 (French, with English summary). MR 1010769
- V. A. Borovikov, Difraktsiya na mnogougol′nikakh i mnogogrannikakh, Izdat. “Nauka”, Moscow, 1966 (Russian). MR 0218058
- V. B. Poruchikov, Metody dinamicheskoĭ teorii uprugosti, “Nauka”, Moscow, 1986 (Russian). MR 846431
- V. A. Dobrushkin, Kraevye zadachi dinamicheskoĭ teorii uprugosti dlya klinovidnykh oblasteĭ, “Nauka i Tekhnika”, Minsk, 1988 (Russian). MR 993072
- I. I. Mel′nikov, Singularities of the solution of a mixed problem for second-order hyperbolic equations in domains with a piecewise-smooth boundary, Uspekhi Mat. Nauk 37 (1982), no. 1(223), 149–150 (Russian). MR 643778
- Nguen Man′Khung, Asymptotic behavior of solutions of the first boundary value problem for strongly hyperbolic systems near a conical point of the domain boundary, Mat. Sb. 190 (1999), no. 7, 103–126 (Russian, with Russian summary); English transl., Sb. Math. 190 (1999), no. 7-8, 1035–1058. MR 1725214, DOI 10.1070/SM1999v190n07ABEH000415
- Vladimir Gilelevič Maz′ja and Jürgen Rossmann, Über die Asymptotik der Lösungen elliptischer Randwertaufgaben in der Umgebung von Kanten, Math. Nachr. 138 (1988), 27–53 (German). MR 975198, DOI 10.1002/mana.19881380103
- Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi, Higher transcendental functions. Vols. I, II, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1953. Based, in part, on notes left by Harry Bateman. MR 0058756
- Lars Hörmander, The analysis of linear partial differential operators. I, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 256, Springer-Verlag, Berlin, 1983. Distribution theory and Fourier analysis. MR 717035, DOI 10.1007/978-3-642-96750-4
- V. G. Maz′ja and B. A. Plamenevskiĭ, Asymptotic behavior of the fundamental solutions of elliptic boundary value problems in domains with conical points, Boundary value problems. Spectral theory (Russian), Probl. Mat. Anal., vol. 7, Leningrad. Univ., Leningrad, 1979, pp. 100–145, 243 (Russian). MR 559106
Bibliographic Information
- A. Kokotov
- Affiliation: Concordia University, Montreal, Canada
- MR Author ID: 252297
- Email: kokotov@online.ru
- B. Plamenevskiĭ
- Affiliation: Department of Physics, St. Petersburg State University, Ulyanovskaya 1, Petrodvorets, St. Petersburg 198504, Russia
- Email: boris.plamenevskij@pobox.spbu.ru
- Received by editor(s): December 1, 2003
- Published electronically: May 2, 2005
- © Copyright 2005 American Mathematical Society
- Journal: St. Petersburg Math. J. 16 (2005), 477-506
- MSC (2000): Primary 35C20, 35L20
- DOI: https://doi.org/10.1090/S1061-0022-05-00862-9
- MathSciNet review: 2083566