Siberian Mathematical Journal

, Volume 49, Issue 5, pp 842–851 | Cite as

Quasielliptic operators and Sobolev type equations

  • G. V. DemidenkoEmail author


We consider a class of matrix quasielliptic operators on the n-dimensional space. For these operators, we establish the isomorphism properties in some special scales of weighted Sobolev spaces. Basing on these properties, we prove the unique solvability of the initial value problem for a class of Sobolev type equations.


quasielliptic operator weighted Sobolev space isomorphism Sobolev type equations 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Sobolev S. L., Introduction to the Theory of Cubature Formulas [in Russian], Nauka, Moscow (1974).Google Scholar
  2. 2.
    Cantor M., “Spaces of functions with asymptotic conditions on ℝn,” Indiana Univ. Math. J., 24, No. 9, 897–902 (1975).zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    McOwen R. C., “The behavior of the Laplacian on weighted Sobolev spaces,” Comm. Pure Appl. Math., 32, No. 6, 783–795 (1979).zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bagirov L. A. and Kondrat’ev V. A., “On elliptic equations in ℝn,” Differentsial’nye Uravneniya, 11, No. 3, 498–504 (1975).zbMATHMathSciNetGoogle Scholar
  5. 5.
    Cantor M., “Some problems of global analysis on asymptotically simple manifolds,” Compositio Math., 38, No. 1, 3–35 (1979).zbMATHMathSciNetGoogle Scholar
  6. 6.
    McOwen R. C., “On elliptic operators in ℝn,” Comm. Partial Differential Equations, 5, No. 9, 913–933 (1980).zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Lockhart R. B., “Fredholm properties of a class of elliptic operators on non-compact manifolds,” Duke Math. J., 48, No. 1, 289–312 (1981).zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Choquet-Bruhat Y. and Christodoulou D., “Elliptic systems in H s, σ spaces on manifolds which are Euclidean at infinity,” Acta Math., 146, No. 1/2, 129–150 (1981).zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Lockhart R. B. and McOwen R. C., “Elliptic differential operators on noncompact manifolds,” Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 12, No. 3, 409–447 (1985).zbMATHMathSciNetGoogle Scholar
  10. 10.
    Demidenko G. V., “On quasielliptic operators in ℝn,” Siberian Math. J., 39, No. 5, 884–893 (1998).CrossRefMathSciNetGoogle Scholar
  11. 11.
    Hile G. N., “Fundamental solutions and mapping properties of semielliptic operators,” Math. Nachr., 279, No. 13–14, 1538–1572 (2006).zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Demidenko G. V., “Isomorphic properties of one class of differential operators and their applications,” Siberian Math. J., 42, No. 5, 865–883 (2001).CrossRefMathSciNetGoogle Scholar
  13. 13.
    Demidenko G. V., “On one class of matrix differential operators,” Siberian Math. J., 45, No. 1, 86–99 (2004).CrossRefMathSciNetGoogle Scholar
  14. 14.
    Demidenko G. V., “Mapping properties of quasielliptic operators and applications,” Int. J. Dynamical Systems Differential Equations, 1, No. 1, 58–67 (2007).CrossRefGoogle Scholar
  15. 15.
    Volevich L. R., “Local properties of solutions to quasielliptic systems,” Mat. Sb., 59, No. 3, 3–52 (1962).MathSciNetGoogle Scholar
  16. 16.
    Lions J.-L., Optimal Control of Systems Governed by Partial Differential Equations [Russian translation], Mir, Moscow (1972).Google Scholar
  17. 17.
    Demidenko G. V., “On weighted Sobolev spaces and integral operators determined by quasielliptic equations,” Russ. Acad. Sci., Dokl., Math., 49, No. 1, 113–118 (1994).MathSciNetGoogle Scholar
  18. 18.
    Kudryavtsev L. D., “Embedding theorems for functions on the whole space or a half-space,” I: Mat. Sb., 69, No. 4, 616–639 (1966); II: Mat. Sb., 70, No. 1, 3–35 (1966).Google Scholar
  19. 19.
    Nirenberg L. and Walker H. F., “The null spaces of elliptic partial differential operators in ℝn,” J. Math. Anal. Appl., 42, No. 2, 271–301 (1973).zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Cantor M., “Elliptic operators and the decomposition of tensor fields,” Bull. Amer. Math. Soc., 5, No. 3, 235–262 (1981).zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Sobolev S. L., Selected Works. Vol. 1; 2 [in Russian], Filial “Geo” Izdat. Sibirsk. Otdel. RAN, Novosibirsk (2003; 2006).Google Scholar
  22. 22.
    Demidenko G. V. and Uspenskiĭ S. V., Partial Differential Equations and Systems Not Solvable with Respect to the Highest-Order Derivative, Marcel Dekker, New York; Basel (2003).zbMATHGoogle Scholar
  23. 23.
    Uspenskiĭ S. V., “On the representation of functions defined by a class of hypoelliptic operators,” Proc. Steklov Inst. Math., 117, 343–352 (1974).Google Scholar
  24. 24.
    Lizorkin P. I., “Generalized Liouville differentiation and the multiplier method in the theory of embeddings of classes of differentiable functions,” Trudy Mat. Inst. Steklov, 105, 89–167 (1969).zbMATHMathSciNetGoogle Scholar
  25. 25.
    Hardy G. H., Littlewood D. E., and Pólya G., Inequalities, Cambridge University Press, Cambridge (UK) etc. (1988).zbMATHGoogle Scholar
  26. 26.
    Sobolev S. L., “On a new problem of mathematical physics,” Izv. Akad. Nauk SSSR Ser. Mat., 18, No. 1, 3–50 (1954).MathSciNetGoogle Scholar
  27. 27.
    Friedrichs K. O., “Symmetric hyperbolic linear differential equations,” Comm. Pure Appl. Math., 7, No. 2, 345–392 (1954).zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2008

Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsNovosibirskRussia

Personalised recommendations