Skip to main content
SpringerLink
Log in

Multicomponent conjugation problems and auxiliary abstract boundary-value problems

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  1. Yu. Sh. Abramov, Variational Methods in the Theory of Operator Sheaves. Spectral Optimization [in Russian], Leningrad Univ. (1983).

  2. M. S. Agranovich, “Remarks on potential spaces and Besov spaces in Lipschitz domains and on Whitney arrays on their boundaries,” Russ. J. Math. Phys., 15, No. 2, 146–155 (2008).

    Article  MATH  MathSciNet  Google Scholar 

  3. M. S. Agranovich, B. A. Amosov, and M. Levitin, “Spectral problems for the Lam´e system with spectral parameter in boundary conditions on smooth and nonsmooth boundaries,” Russ. J. Math. Phys., 6, No. 3, 247–281 (1999).

    MATH  MathSciNet  Google Scholar 

  4. M. S. Agranovich, B. Z. Katsenelenbaum, A. N. Sivov, and N. N. Voitovich, Generalized Method of Eigenoscillations in Diffraction Theory, Wiley, Berlin (1999).

    MATH  Google Scholar 

  5. M. S. Agranovich and R. Mennicken, “Spectral value problems for the Helmholtz equation with spectral parameter in boundary condition on a nonsmooth surface,” Sb. Math., 190, No. 1, 29–69 (1998).

    Article  MathSciNet  Google Scholar 

  6. O. A. Andronova and N. D. Kopachevsky, “Mixed and spectral problems with surface dissipation of energy,” in: Proc. of Ukrainian Sci. Conf. of Young Scientists and Students on Differential Equations and Applications (in honor of Ya. B. Lopatinskii), Donetsk (Ukraine), December 6-7 (2006), pp. 12–13.

  7. O. Andronova and N. Kopachevsky, “Operator approach to dynamic systems with surface dissipation of energy,” in: Int. Conf. “Modern Analysis and Applications” dedicated to the centenary of Mark Krein, Odessa, Ukraine, April 9-14 (2007), p. 11.

  8. N. K. Askerov, S. G. Krein, and G. I. Laptev, “Oscillations of a viscous liquid and the associated operational equations,” Funkts. Anal. Prilozh., 2, No. 2, 21–31 (1968).

    MATH  MathSciNet  Google Scholar 

  9. T. Ya. Azizov and I. S. Iokhvidov, Foundations of the Theory of Linear Operators in Spaces with Indefinite Metric [in Russian], Nauka, Moscow (1986).

    MATH  Google Scholar 

  10. T. Ya. Azizov and N. D. Kopachevsky, “On the basis property of the system of eigen- and associated elements of the S. G. Krein problem on normal oscillations of a viscous fluid in an open vessel,” in: Spectral and Evolutionary Problems, 3 (1994), pp. 38–39.

  11. V. V. Barkovskii, “A decomposition by eigenfunctions and eigenvectors related to general elliptic problems with an eigenvalue in the boundary condition,” Ukr. Mat. Zh., 19, No. 1, (1967).

  12. B. V. Bazalii and S. P. Degtyarev, “On the Stefan problem with kinetic and classical conditions at a free boundary,” Ukr. Mat. Zh., 44, No. 2, 155–166 (1992).

    Article  MathSciNet  Google Scholar 

  13. J. Below and G. François, “Spectral asymptotics for the Laplacian under an eigenvalue-dependent boundary condition,” Bull. Belgian Math. Soc. Simon Stevin, 12, No. 4, 505–519 (2005).

    MATH  Google Scholar 

  14. I. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems [in Russian], Ada, Krakov (1999).

    MATH  Google Scholar 

  15. I. Chueshov, M. Eller, and I. Lasiecka, “Finite dimension of the attractor for a semilinear wave equation with nonlinear boundary dissipation,” Commun. PDE, 29, No 1–12, 1847-1876 (2004).

    MATH  MathSciNet  Google Scholar 

  16. J. Ercolano and M. Schechter, “Spectral theory for operators generated by elliptic boundary-value problems with eigenvalue parameter in boundary conditions,” Commun. Pure Appl. Math., 18, 83–105 (1965).

    Article  MathSciNet  Google Scholar 

  17. S. F. Feshchenko, I. A. Lukovskii, B. I. Rabinovich, and L. V. Dokuchaev, Methods of Determination of Adjoint Fluid Masses in Moving Domains [in Russian], Naukova Dumka, Kiev, 216–224 (1969).

    Google Scholar 

  18. E. Gagliardo, “Caratterizzazioni delle trace sulla frontiera relative ad alaine classi di funzioni in n variabili,” Rend. Semin. Mat. Univ. Padova, 27, 284–305 (1957).

    MATH  MathSciNet  Google Scholar 

  19. I. C. Gokhberg and M. G. Krein, Introduction to the Theory of Linear Non-Self-Adjoint Operators, Transl. Math. Monogr., 18, Amer. Math. Soc., Providence, Rhode Island (1969).

    Google Scholar 

  20. V. I. Gorbachuk and M. L. Gorbachuk, Boundary-Value Problems for Differential Operator Equations [in Russian], Naukova Dumka, Kiev (1984).

    MATH  Google Scholar 

  21. V. A. Grinshtein and N. D. Kopachevsky, On the Spectra of Bounded Self-Adjoint Operators [in Russian], preprint (1989).

  22. V. A. Grinshtein and N. D Kopachevsky, “On the p-basicity of a system of eigenelements of a selfadjoint operator-valued function,” Proc. XV All-Union School on Operator Theory in Functional Spaces, Ul’yanovsk, September 5–12, 1990, 1, (1990), p. 12.

  23. N. D. Kopachevsky, “On the basis properties of a system of eigenvectors and associated vectors of a self-adjoint operator sheaf I − λA − λ −1 B,” Funkts. Anal. Prilozh., 15, No. 2, 77–78 (1981).

    Article  Google Scholar 

  24. N. D. Kopachevsky, “On the p-basis property of the system of root vectors of a self-adjoint operator pensil I − λA − λ −1 B,” in: Functional Analysis and Applied Mathematics [in Russian], Naukova Dumka, Kiev (1982), pp. 43–55.

    Google Scholar 

  25. N. D. Kopachevsky, “On the abstract Green formula for a triplet of Hilbert spaces and its application to the Stokes problem,” Taurida Bull. Inform. Math., No. 2, 52–80 (2004).

    Google Scholar 

  26. N. D. Kopachevsky, “On abstract boundary value problems with surface dissipation of energy,” in: Proc. Int. Conf. “Analysis and Partial Differential Equations” (in honor of Prof. B. Bojarsky), Bedlevo, Poland, June 18–24, (2006), p. 23.

  27. N. D. Kopachevsky, “The abstract Green formula for mixed boundary-value problems,” Uch. Zap. Taurida Natl. Univ. Ser. Mat. Mekh. Inform. Kibern., 20(59), No. 2, 3–12 (2007).

    Google Scholar 

  28. N. D. Kopachevsky and S. G. Krein, Operator Approach to Linear Problems of Hydrodynamics. Vol. 1. Self-Adjoint Problems for Ideal Fluids, Birkhäuser, Basel–Boston–Berlin (2001).

    Google Scholar 

  29. N. D. Kopachevsky and S. G. Krein, Operator Approach to Linear Problems of Hydrodynamics. Vol. 2. Non-Self-Adjoint Problems for Viscous Fluids. Birkhäuser, Basel–Boston–Berlin (2003).

    Google Scholar 

  30. N. D. Kopachevsky and S. G. Krein, “The abstract Green formula for a triplet of Hilbert spaces, abstract initial-value and spectral problem,” Ukr. Math. Vist., 1, No. 1, 69–97 (2004).

    Google Scholar 

  31. N. D. Kopachevsky, S. G. Krein, and N. Z. Kan, Operator Methods in Linear Hydrodynamics. Evolutionary and Spectral Problems [in Russian], Nauka, Moscow (1989).

    Google Scholar 

  32. N. D. Kopachevsky and O. I. Nemirskaya, On the p-Basis Property of the System of Eigenlements of Self-Adjoint Operator-Valued Functions [in Russian], preprint, Simferopol Univ. (1992).

  33. N. D. Kopachevsky and P. A. Starkov, “Operator approach to transmission problems for Helmholtz equation,” in: Ukr. Congr. Math., Nonlinear PDEs, Kyiv, August 22–28 (2001), p. 68.

  34. N. D. Kopachevsky and P. A. Starkov, “Abstract Green’s formula and transmission problems,” in: Int. Conf. on Functional Analysis and Its Applications dedicated to the 110th anniversary of S. Banach, Lviv, Ukraine, May 28–31 (2002), pp. 112–113.

  35. N. D. Kopachevsky, P. A. Starkov, and V. I. Voytitsky, “Abstract Green’s formula and spectral transmission problems,” Int. Conf. Nonlinear PDEs, Yalta, Crimea, Ukraine, September 10–15 (2007), pp. 40–41.

  36. N. D. Kopachevsky and V. I. Voytitsky, “On the modified spectral Stefan problem and its abstract generalizations,” in: Operator Theory: Advances and Applications, 191, Birkhäuser, Basel (2009), pp. 373–386.

    Google Scholar 

  37. S. G. Krein (ed.), Functional Analysis [in Russian], Nauka, Moscow (1972).

    Google Scholar 

  38. S. G. Krein, “On oscillations of a viscous fluid in a vessel,” Dokl. Akad. Nauk SSSR 159, No. 2, 262–265 (1964).

    MathSciNet  Google Scholar 

  39. S. G. Krein and G. I. Laptev, “Motion of a viscous liquid in an open vessel,” Funkts. Anal. Prilozh., 2, No. 1, 40–50 (1968).

    MATH  MathSciNet  Google Scholar 

  40. V. E. Liantse and O. G. Storozh, Methods of the Theory of Nonbounded Operators [in Russian], Naukova Dumka, Kiev (1983).

    Google Scholar 

  41. J.-L. Lions and E. Magenes, Non-Homogeneous Boundary-Value Problems and Applications, Springer-Verlag, Berlin–New York (1972).

    Google Scholar 

  42. A. S. Markus, Introduction to the Spectral Theory of Polynomial Operator Sheaves [in Russian], Shtiintsa, Kishinev (1986).

    Google Scholar 

  43. A. S. Marcus and V. I. Matsaev, “On the basis property for a certain part of the eigenvectors and associated vectors of the self-adjoint operator sheaf,” Math. USSR Sb., 61, No. 2, 289–307 (1988).

    Article  MathSciNet  Google Scholar 

  44. A. S. Marcus and V. I. Matsaev, “Comparison theorems for spectra of linear operators and spectral asymptotics,” Tr. Mosk. Mat. Obshch., 45, 133–181 (1982).

    Google Scholar 

  45. A. S. Marcus and V. I. Matsaev, “A theorem on comparison of spectra and the spectral asymptotics for Keldysh sheaves,” Math. USSR. Sb., 61, No. 2, 289–307 (1988).

    Article  MathSciNet  Google Scholar 

  46. J.-P. Oben, “Approximate solution of elliptic boundary-value problems,” Sov. Math. Dokl., 38, No. 2, 327–331 (1989).

    Google Scholar 

  47. B. V. Pal’tzev, “On a mixed problem with a nonuniform boundary conditions for second-ordered elliptic equations with a parameter in Lipschitz domains,” Math. Sb., 187, No. 4, 59–116 (1996).

    Google Scholar 

  48. E. V. Radkevich, “On conditions for the existence of classical solutions of the modified Stefan problem (the Gibbs–Thompson law),” Math. USSR Sb., 75, No. 1, 221–246 (1993).

    MathSciNet  Google Scholar 

  49. V. S. Rychkov, “On restrictions and extensions of the Besov and Triebel–Lizorkin spaces with respect to Lipschitz domains,” J. London Math. Soc., 60, 237–257 (1999).

    Article  MathSciNet  Google Scholar 

  50. R. Showalter, “Hilbert space methods for partial differential equations,” Electron. J. Differ. Eqs. (1994).

  51. V. I. Smirnov, A Course in Higher Mathematics. Part V, H. Deutsch, Frankfurt-am-Main (1991).

    Google Scholar 

  52. P. A. Starkov, “An operator approach to conjugation problems,” Uch. Zap. Taurida Natl. Univ. Ser. Mat. Mekh. Inform. Kibern., 15 (54), No. 1, 58–62 (2002).

    MathSciNet  Google Scholar 

  53. P. A. Starkov, “A case of a general position for an operator sheaf arising in studying of conjugation problems,” Uch. Zap. Taurida Natl. Univ. Ser. Mat. Mekh. Inform. Kibern., 15 (54), No. 2, 82–88 (2002).

    Google Scholar 

  54. P. A. Starkov, “The study of conjugation problems for exceptional values of a fixed parameter,” Uch. Zap. Taurida Natl. Univ. Ser. Mat. Mekh. Inform. Kibern., 16 (55), No. 2, 81–89 (2003).

    Google Scholar 

  55. P. A. Starkov, “On the basis property of a system of eigenelements in conjugation problems,” Taurida Bull. Inform. Math., 1, 118–131 (2003).

    Google Scholar 

  56. P. A. Starkov, “Examples of multicomponent conjugation problems,” Uch. Zap. Taurida Natl. Univ. Ser. Mat. Mekh. Inform. Kibern., 18 (57), No. 1, 89–94 (2005).

    MathSciNet  Google Scholar 

  57. L. R. Volevich and S. G. Gindikin, Generalized Functions and Convolution Equations [in Russian], Nauka, Moscow (1994).

    Google Scholar 

  58. V. I. Voytitsky, “Abstract spectral Stefan problem,” Uch. Zap. Taurida Natl. Univ. Ser. Mat. Mekh. Inform. Kibern., 19 (58), No. 2, 20–28 (2006).

    Google Scholar 

  59. V. I. Voytitsky, “On spectral problems generated by the Stefan problem with the Gibbs–Thompson conditions,” Nonlin. Boundary-Value Problems, 17, 31–49 (2007).

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Mathematics and Informatics, Chair of Mathematical Analysis, Taurida National V. I. Vernadsky University, Simferopol, Ukraine

    V. I. Voititsky, N. D. Kopachevsky & P. A. Starkov

Authors
  1. V. I. Voititsky
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. N. D. Kopachevsky
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. P. A. Starkov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. I. Voititsky.

Additional information

Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 34, Proceedings of KROMSH, 2009.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Voititsky, V.I., Kopachevsky, N.D. & Starkov, P.A. Multicomponent conjugation problems and auxiliary abstract boundary-value problems. J Math Sci 170, 131–172 (2010). https://doi.org/10.1007/s10958-010-0078-8

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-010-0078-8

Keywords

Search

Navigation