Potential Analysis

, Volume 35, Issue 3, pp 201–222

Regular Degenerate Separable Differential Operators and Applications

Article

Abstract

The boundary value problems for differential-operator equations with variable coefficients, degenerated on all boundary are studied. Several conditions for the separability, fredholmness and resolvent estimates in Lp-spaces are given. In applications degenerate Cauchy problem for parabolic equation, boundary value problems for degenerate partial differential equations and systems of degenerate elliptic equations on cylindrical domain are studied.

Keywords

Degenerate differential-operator equations Semigroups of operators Banach-valued function spaces Separability Fredholmness Operator-valued Fourier multipliers Interpolation of Banach spaces 

Mathematics Subject Classifications (2010)

34G10 35J25 35J70 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Agmon, S.: On the eigenfunctions and on the eigenvalues of general elliptic boundary value problems. Commun. Pure Appl. Math. 15, 119–147 (1962)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Ashyralyev, A.: On well-posedeness of the nonlocal boundary value problem for elliptic equations. Numer. Funct. Anal. Optim. 24(1 & 2), 1–15 (2003)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Amann, H.: Linear and Quasi-Linear Equations,1. Birkhauser, Basel (1995)Google Scholar
  4. 4.
    Aubin, J.P.: Abstract boundary-value operators and their adjoint. Rend. Semin. Mat. Univ. Padova 43, 1–33 (1970)MathSciNetGoogle Scholar
  5. 5.
    Agarwal, R., O’ Regan, D., Shakhmurov, V.B.: Separable anisotropic differential operators in weighted abstract spaces and applications. J. Math. Anal. Appl. 338, 970–983 (2008)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Besov, O.V., Ilin, V.P., Nikolskii, S.M.: Integral Representations of Functions and Embedding Theorems. Nauka, Moscow (1975)Google Scholar
  7. 7.
    Burkholder, D.L.: A geometrical conditions that implies the existence certain singular integral of Banach space-valued functions. In: Proc. Conf. Harmonic Analysis in Honor of Antonu Zigmund, Chicago, 1981, pp. 270–286. Wads Worth, Belmont (1983)Google Scholar
  8. 8.
    Clement, Ph., de Pagter, B., Sukochev, F.A., Witvliet, H.: Schauder decomposition and multiplier theorems. Stud. Math. 138, 135–163 (2000)Google Scholar
  9. 9.
    Dore, C., Yakubov, S.: Semigroup estimates and non coercive boundary value problems. Semigroup Forum 60, 93–121 (2000)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Dore, G., Venni, A.: On the closedness of the sum of two closed operators. Math. Z. 196, 189–201 (1987)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Denk, R., Hieber, M., Prüss, J.: R-boundedness, fourier multipliers and problems of elliptic and parabolic type. Mem. Am. Math. Soc. 166(788), Providence RI (2003)Google Scholar
  12. 12.
    Favini, A., Shakhmurov, V.B., Yakubov, Y.: Regular boundary value problems for complete second order elliptic differential-operator equations in UMD Banach spaces. Semigroup form. 79(1), 22–54 (2009)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Grisvard, P.: Commutative’ de deux foncteurs d’interpolation et applications. J. Math. Pures Appl. 45(9), 143–290 (1966)MathSciNetGoogle Scholar
  14. 14.
    Haller, R., Heck, H., Noll, A.: Mikhlin’s theorem for operator-valued Fourier multipliers in n variables. Math. Nachr. 244, 110–130 (2002)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Goldstain, J.A.: Semigroups of Linear Operators and Applications. Oxford Mathematical Monographs. Oxford University Press, Clarendon Press, New York and Oxford (1985)Google Scholar
  16. 16.
    Krein, S.G.: Linear Differential Equations in Banach Space. American Mathematical Society, Providence (1971)Google Scholar
  17. 17.
    Komatsu, H.: Fractional powers of operators. Pac. J. Math. 19, 285–346 (1966)MathSciNetGoogle Scholar
  18. 18.
    Lunardi, A.: Analytic Semigroups and Optimal Regularity in Parabolic Problems. Birkhauser (2003)Google Scholar
  19. 19.
    Lizorkin, P.I.: \(\left( L_{p},L_{q}\right) \)-multiplicators of fourier integrals. Dokl. Akad. Nauk SSSR 152(4), 808–811 (1963)MathSciNetGoogle Scholar
  20. 20.
    Lamberton, D.: Equations d’evolution linéaires associées à des semi-groupes de contractions dans les espaces L p. J. Funct. Anal. 72, 252–262 (1987)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Nazarov, S.A., Plammenevskii, B.A.: Elliptic Problems in Domains with Piecewise Smooth Boundaries. Walter de Gruyter, New York (1994)CrossRefGoogle Scholar
  22. 22.
    Shklyar, A.Ya.: Complete Second Order Linear Differential Equations in Hilbert Spaces. Birkhauser Verlak, Basel (1997)Google Scholar
  23. 23.
    Sobolevskii, P.E.: Coerciveness inequalities for abstract parabolic equations. Dokl. Akad. Nauk SSSR 57(1), 27–40 (1964)Google Scholar
  24. 24.
    Shakhmurov, V.B.: Theorems about of compact embedding and applications. Dokl. Akad. Nauk SSSR 241(6), 1285–1288 (1978)MathSciNetGoogle Scholar
  25. 25.
    Shakhmurov, V.B.: Theorems on compactness of embedding in weighted anisotropic spaces, and their applications. Dokl. Akad. Nauk SSSR 291(6), 612–616 (1986)MathSciNetGoogle Scholar
  26. 26.
    Shakhmurov, V.B.: Imbedding theorems and their applications to degenerate equations. Diff. Equ. 24(4), 475–482 (1988)MathSciNetGoogle Scholar
  27. 27.
    Shakhmurov, V.B.: Coercive boundary value problems for regular degenerate differential-operator equations. J. Math. Anal. Appl. 292(2), 605–620 (2004)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Shakhmurov, V.B.: Embedding theorems and maximal regular differential operator equations in Banach-valued function spaces. J. Inequal. Appl. 2(4), 329–345 (2005)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Shakhmurov, V.B.: Separable anisotropic differential operators and applications. J. Math. Anal. Appl. 327(2), 1182–1201 (2006)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. North-Holland, Amsterdam (1978)Google Scholar
  31. 31.
    Weis, L.: Operator-valued Fourier multiplier theorems and maximal L p regularity. Math. Ann. 319, 735–758 (2001)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Yakubov, S.: Completeness of Root Functions of Regular Differential Operators. Longman, Scientific and Technical, New York (1994)Google Scholar
  33. 33.
    Yakubov, S.: A nonlocal boundary value problem for elliptic differential-operator equations and applications. Integr. Equ. Oper. Theory 35, 485–506 (1999)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Yakubov, S., Yakubov, Ya.: Differential-Operator Equations. Ordinary and Partial Differential Equations. Chapman and Hall/CRC, Boca Raton (2000)Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Department of Electronics and CommunicationOkan UniversityIstanbulTurkey

Personalised recommendations