On the Behavior at Infinity of Solutions of Elliptic Systems with a Finite Energy Integral

  • V. A. Kondratiev
  • O. A. Oleinik

Abstract

Asymptotic behavior in a neighborhood of infinity is studied in this paper for solutions of an elliptic system of order 2m with constant complex coefficients. It is supposed that the weighted Dirichlet integral is bounded. Our considerations include solutions with finite energy for the system of linear elasticity (see Theorem 3). A class of solutions periodic in some independent variables is also studied in this paper (the E. Sanchez-Palencia problem).

Keywords

Summing 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    J. Serrin & H. Weinberger, Isolated singularities of linear elliptic equations. Amer. Math. Journ., 1966, v. 88, Nr. 1, p. 258–272.CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    V. A. Kondratiev & O. A. Oleinik, Sur un probléme de E. Sanchez-Palencia. C. R. Acad. Sci. Paris, 1984, v. 299, ser. 1, Nr. 15, p. 745–748.MATHMathSciNetGoogle Scholar
  3. 3.
    V. A. Kondratiev & O. A. Oleinik, On periodic in the time solutions of a second order parabolic equation in exterior domains. Vestnik of the Moscow University, Math. Mech., ser. 1, 1985, Nr. 4, p. 38–47.Google Scholar
  4. 4.
    B. R. Wainberg, On solutions of elliptic equations with constant coefficients and a right hand side growing at the infinity. Vestnik of the Moscow University. Math. Mech., ser. 1, 1968, Nr. 1, p. 41–48.Google Scholar
  5. 5.
    Ja. B. Lopatinsky, The behaviour at infinity of solutions of systems of differential equations of the elliptic type. Dokl. AN Ukr. SSR, 1959, Nr. 9, p. 931–935.Google Scholar
  6. 6.
    S. L. Sobolev, Introduction to the theory of cubature formulas. Moscow, Nauka, 1974.Google Scholar
  7. 7.
    E. M. Landis & G. P. Panasenko, On a variant of the Phragmen-Lindelöf type theorem for elliptic equations with coefficients periodic in all variables except one. Trudi seminara imeni I. G. Petrovsky, 1979, v. 5, p. 105–136.MathSciNetGoogle Scholar
  8. 8.
    O. A. Oleinik & G. A. Yosifian, On the behavior at infinity of solutions of second order elliptic equations in domains with noncompact boundary. Matem. Sbórnik, 1980, v. 112, Nr. 4, p. 588–610.Google Scholar
  9. 9.
    O. A. Oleinik & G. A. Yosifian, On the asymptotic behavior at infinity of solutions in linear elasticity. Archive Rational Mech. and Analysis, 1982, v. 78, Nr. 1, p. 29–53.MATHADSMathSciNetGoogle Scholar
  10. 10.
    A. Douglis & L. Nirenberg, Interior estimates for elliptic systems of partial differential equations. Comm. Pure Appl. Math., 1955, v. 8, p. 503–538.CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Ja. B. Lopatinsky, Fundamental solutions of a system of differential equations of elliptic type. Ukr. mat. journal, 1951, v. 3, Nr. 1, p. 3–38.Google Scholar
  12. 12.
    L. Bers, F. John, & M. Schechter, Partial differential equations, Interscience publishers, New York, 1964.MATHGoogle Scholar
  13. 13.
    I. M. Gelfand & G. E. shilov, Generalized functions, v. 3, Moscow, 1958.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1989

Authors and Affiliations

  • V. A. Kondratiev
    • 1
  • O. A. Oleinik
    • 1
  1. 1.Moscow UniversityRussia

Personalised recommendations