Remote Access Transactions of the Moscow Mathematical Society

Transactions of the Moscow Mathematical Society

ISSN 1547-738X(online) ISSN 0077-1554(print)

 
 

 

Neumann problem in angular regions with periodic and parabolic perturbations of the boundary


Author: S. A. Nazarov
Translated by: O. A. Khleborodova
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 69 (2008).
Journal: Trans. Moscow Math. Soc. 2008, 153-208
MSC (2000): Primary 35J25; Secondary 35C20
DOI: https://doi.org/10.1090/S0077-1554-08-00173-8
Published electronically: December 24, 2008
MathSciNet review: 2549447
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We construct and prove asymptotic expansions at infinity for solutions of Neumann problems and matching problems for systems of second order differential equations in regions with corner outlets to infinity. Outside some disc the regions are either periodic or deformed by parabolic inclusions. In addition to logarithmic-polynomial solutions, the asymptotic expansions contain components of the type of boundary layer that either exponentially decay away from the boundary or are localized inside the parabolic inclusions. Operators of the problems become Fredholm operators, and remainders in asymptotic expansions are estimated in scales of function spaces with norms determined by double weight factors and their step distribution. We consider also other types of problems which allow us to apply the developed methods for reduction to a model problem in a sector and for recovery of properties of remainders near perturbed boundary; in particular, we consider the matching problem in regions with irregular points of peak-like inclusion type.


References [Enhancements On Off] (What's this?)

  • 1. Nečas J. Les méthodes directes en théorie des équations elliptiques. Masson, Paris; Prague, 1967. MR 0227584 (37:3168)
  • 2. S. A. Nazarov, Self-adjoint elliptic boundary. Polynomial property and formally positive operators. Problemy Mat. Analiza, no. 16, SPb. Univ., St-Petersburg, 1997, pp. 167-192; English transl., J. Math. Sci. (New York) 92 (1998), no. 6, 4338-4353. MR 1668363 (99h:00043); MR 1668418 (2000a:35042)
  • 3. -, Polynomial property of selfadjoint elliptic boundary value problems, and the algebraic description of their attributes. Uspekhi Mat. Nauk 54 (1999), no. 5(329), 77-142; English transl., Russian Math. Surveys 54 (1999), no. 5, 947-1014. MR 1741662 (2001k:35073)
  • 4. E. Sanchez-Palencia and P. Suquet, Friction and homogenization of a boundary. Free Boundary Problems: Theory and Applications. A.Fasano, M.Primicerio (eds.). Pitman, London, 1983, pp. 561-571. MR 714936 (86b:73036)
  • 5. M. Lobo and M. Pérez, Local problems for vibrating systems with concentrated masses: a review. C.R.Mecanique 331 (2003), 303-317.
  • 6. S. A. Nazarov, Binomial asymptotic behavior of solutions of spectral problems with singular perturbations. Mat. Sb. 181 (1990), no. 3, 291-320; English transl., Math. USSR-Sb. 69 (1991), no. 2, 307-340. MR 1049991 (91d:35160)
  • 7. Y. Amirat, G. A. Chechkin, and R. R. Gadyl'shin, Asymptotics of simple eigenvalues and eigenfunctions for the Laplace operator in a domain with oscillating boundary, J. Comp. Math. Math. Phys. 46 (2006), no. 1, 102-115. MR 2239730 (2007d:35009)
  • 8. A. M. Il'in, Matching of asymptotic expansions of solutions of boundary value problems. ``Nauka'', Moscow, 1989; English transl., Amer. Math. Soc., Providence, RI, 1992. MR 1182791 (93g:35016)
  • 9. W. G. Mazja, S. A. Nazarov, and B. A. Plamenewski, Asymptotische Theorie elliptischer Randwertaufgaben in singulär gestörten Gebieten. 1. Akademie-Verlag, Berlin, 1991; English transl., V. Maz'ya, S. Nazarov, and B. Plamenevskij, Asymptotic theory of elliptic boundary value problems in singularly perturbed domains. Vol.1, Birkhäuser, Basel, 2000. MR 1101139 (92g:35059)
  • 10. V. A. Kondrat'ev, Boundary value problems for elliptic equations in domains with conical or angular points. Trudy Moskov. Mat. Obshch. 16 (1967), 209-292. (Russian) MR 0226187 (37:1777)
  • 11. V. G. Maz'ja and B. A. Plamenevskiĭ, The coefficients in the asymptotics of solutions of elliptic boundary value problems with conical points. Math. Nachr. 76 (1977), 29-60. (Russian) MR 0601608 (58:29176)
  • 12. -, The coefficients in the asymptotic expansion of the solutions of elliptic boundary value problems near an edge. Dokl. Akad. Nauk SSSR 229 (1976), no. 1, 33-36. (Russian) MR 0407446 (53:11221)
  • 13. S. A. Nazarov and B. A. Plamenevsky, Elliptic problems in domains with piecewise smooth boundaries. Walter de Gruyter, Berlin-New York, 1994. MR 1283387 (95h:35001)
  • 14. V. A. Kozlov, V. G. Maz'ya, and J. Rossmann, Elliptic boundary value problems in domains with point singularities. Amer. Math. Soc., Providence, RI, 1997. MR 1469972 (98f:35038)
  • 15. S. A. Nazarov, Asymptotic behavior of the solution of the Dirichlet problem in an angular domain with a periodically changing boundary. Mat. Zametki 49 (1991), no. 5, 86-96; English transl., Math. Notes 49 (1991), no. 5-6, 502-509. MR 1137177 (92g:35062)
  • 16. -, The Dirichlet problem for an elliptic system with periodic coefficients in a corner domain. Vestnik Leningrad. Univ. Mat. Mekh. Astronom. 1990, vyp. 1, 32-35; English transl., Vestnik Leningrad Univ. Math. 23 (1990), no. 1, 33-35. MR 1098480 (91m:35079)
  • 17. -, Asymptotic expansion of the solution of the Dirichlet problem for an equation with rapidly oscillating coefficients in a rectangle. Mat. Sb. 182 (1991), no. 5, 692-722; English transl., Math. USSR-Sb. 73 (1992), no. 1, 79-110. MR 1124104 (92h:35038)
  • 18. -, Asymptotics at infinity of the solution to the Dirichlet problem for a system of equations with periodic coefficients in an angular domain. Russian J. Math. Phys. 3 (1995), no. 3. 297-326. MR 1370627 (97g:35034)
  • 19. S. A. Nazarov and A. S. Slutskiĭ, Asymptotic behavior of solutions of boundary value problems for an equation with rapidly oscillating coefficients in a domain with a small cavity. Mat. Sb. 189 (1998), no. 9, 107-142; English transl., Sb. Math. 189 (1998), no. 9-10, 1385-1422. MR 1680848 (2000a:35014)
  • 20. F. Blanc and S. A. Nazarov, Asymptotics of solutions to the Poisson problem in a perforated domain with corners. J.Math. Pures. Appl. 76 (1997), no. 10, 893-911. MR 1489944 (98h:35017)
  • 21. G. Caloz, M. Costabel, M. Dauge, and G. Vial, Asymptotic expansion of the solution of an interface problem in a polygonal domain with thin layer. Asymptotic Analysis. 50 (2006), no. 1, 121-173. MR 2286939
  • 22. D. Leguillon and E. Sanchez-Palencia, Computation of singular solutions in elliptic problems and elasticity. Masson, Paris; Wiley, New York, 1987. MR 995254 (90m:73015)
  • 23. A. B. Movchan and S. A. Nazarov, Asymptotic behavior of the stress-strained state near sharp inclusions. Dokl. Akad. Nauk SSSR 290 (1986), no. 1, 48-51; English transl., Soviet Phys. Dokl. 31 (1986), 772-774. MR 857872 (88c:73019)
  • 24. -, Stress-deformed states in the vertex of a sharp inclusion. Mekh. Tverdogo Tela, 1986, no. 3, 155-163. (Russian)
  • 25. -, Asymptotics of a stress-deformed state near a spatial pique-like inclusion. Mekh. Kompoz. Materialov, 1985, no. 5, pp. 792-800. (Russian)
  • 26. -, Asymptotics of the solution to the Neumann problem in a domain with singular point of peak exterior type. Russian J. Math. Phys. 4 (1996), no. 2, 217-250. MR 1414884 (2000d:35044)
  • 27. V. G. Maz'ya, S. A. Nazarov, and B. A. Plamenevskiĭ, Elliptic boundary value problems in domains of the type of the exterior of a cusp. Linear and nonlinear partial differential equations. Spectral asymptotic behavior, pp. 105-148, Probl. Mat. Anal., 9, Leningrad. Univ., Leningrad, 1984. (Russian) MR 772047 (86g:35063)
  • 28. S. A. Nazarov, Estimates near an edge for the solution of the Neumann problem for an elliptic system. Vestnik Leningrad. Univ. Mat. Mekh. Astronom. 1988, no. 1, 37-42; English transl., Vestnik Leningrad Univ. Math. 21 (1988), no. 1, 52-59. MR 946462 (89h:35108)
  • 29. 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., Vol. 1, 1990, pp. 174-211. (Russian) MR 1104210 (92e:35065)
  • 30. M. S. Agranovich and M. I. Vishik, Elliptic problems with a parameter and parabolic problems of general type. Uspehi Mat. Nauk 19 (1964), no. 3, 53-161; English transl., Russian Math. Surveys 19 (1964), no. 3, 53-157. MR 0192188 (33:415)
  • 31. V. G. Maz'ja and B. A. Plamenevskiĭ, Weighted spaces with inhomogeneous norms, and boundary value problems in domains with conical points. Elliptische Differentialgleichungen (Rostock, 1977), pp. 161-190, Wilhelm-Pieck-Univ., Rostock, 1978. (Russian) MR 540196 (81e:35045)
  • 32. -, Elliptic boundary value problems on manifolds with singularities. Problems in mathematical analysis, No. 6: Spectral theory, boundary value problems, pp. 85-142, Izdat. Leningrad. Univ., Leningrad, 1977. (Russian) MR 0509430 (58:23025)
  • 33. S. A. Nazarov, Asymptotic behavior with respect to a parameter of the solution to a boundary value problem that is elliptic in the sense of Agranovich-Vishik in a domain with conical points. Boundary value problems. Spectral theory, pp. 146-167, 243-244, Probl. Mat. Anal., 7, Leningrad. Univ., Leningrad, 1979. (Russian) MR 559107 (81a:35030)
  • 34. V. A. Nikishkin, Singularities of the solution of the Dirichlet problem for a second-order equation in the neighborhood of an edge. Vestnik Moskov. Univ. Ser. I, Mat. Mekh. no. 2 (1979), 51-62. (Russian) MR 531648 (80h:35046)
  • 35. V. G. Maz'ya and J. Rossmann, Über die Asymptotik der Lösungen elliptischer Randwertaufgaben in der Umgebung von Kanten. Math. Nachr. 138 (1988), 27-53. MR 975198 (90a:35079)
  • 36. S. A. Nazarov, Nonselfadjoint elliptic problems with the polynomial property in domains possessing cylindrical outlets to infinity. Zap. Nauchn. Sem. St.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 249 (1997), 212-230; English transl., J. Math. Sci. (New York) 101 (2000), no. 5, 3512-3522. MR 1698519 (2001b:35087)
  • 37. -, The Vishik-Lyusternik method for elliptic boundary value problems in regions with conic points. II. Problem in a bounded domain. Sibirsk. Mat. Zh. 22 (1981), no. 5, 132-152; English transl., Siberian Math. J. 22 (1981), 753-769. MR 632823 (83m:35046b)
  • 38. S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic differential equations satisfying general boundary conditions. 2. Comm. Pure Appl. Math. 17 (1964), 35-92. MR 0162050 (28:5252)
  • 39. V. A. Solonnikov, General boundary value problems for systems elliptic in the sense of A. Douglis and L. Nirenberg. I. Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 665-706; II. Trudy Mat. Inst. Steklov. 92 MR 0211070 (35:1952)
  • 40. J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Vol. 1. Dunod, Paris, 1968. MR 0247243 (40:512)
  • 41. O. A. Ladyzhenskaya, The boundary value problems of mathematical physics. Springer-Verlag, New York, 1985. MR 793735 (87f:35001)
  • 42. V. A. Kondrat'ev and O. A. Oleinik, Boundary value problems for a system of elasticity theory in unbounded domains. Korn inequalities. Uspekhi Mat. Nauk 43 (1988), no. 5(263), 55-98; English transl., Russian Math. Surveys 43 (1988), no. 5, 65-119. MR 971465 (89m:35061)
  • 43. Ya. A. Roitberg and Z. G. Sheftel, General boundary-value problems for elliptic equations with discontinuous coefficients. Dokl. Akad. Nauk SSSR 148 (1963), 1034-1037. (Russian) MR 0146507 (26:4029)

Similar Articles

Retrieve articles in Transactions of the Moscow Mathematical Society with MSC (2000): 35J25, 35C20

Retrieve articles in all journals with MSC (2000): 35J25, 35C20


Additional Information

S. A. Nazarov
Affiliation: St. Petersburg Branch, Institute of Machine Behavior, Russian Academy of Sciences, St. Petersburg, Russia
Email: serna@snark.ipme.ru

DOI: https://doi.org/10.1090/S0077-1554-08-00173-8
Published electronically: December 24, 2008
Additional Notes: The work was financially supported by the Netherlands Organization for Scientific Research (NWO) and the Russian Fund for Scientific Research (RFFI), joint project 047.017.020.
Article copyright: © Copyright 2008 American Mathematical Society

American Mathematical Society