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Siberian Mathematical Journal

, Volume 11, Issue 1, pp 33–57 | Cite as

Generalized solutions of elliptic equations of second order with coefficients from spaces with mixed norm

  • A. Kh. Gudiev
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Keywords

Generalize Solution Elliptic Equation Mixed Norm 
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Literature Cited

  1. 1.
    S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics [in Russian], Izd. SO AN SSSR (1962).Google Scholar
  2. 2.
    O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Equations of Elliptic Type [in Russian], Nauka, Moscow (1964).Google Scholar
  3. 3.
    A. D. Aleksandrov, “Certain estimates concerning the Dirichlet problem,” Dokl. Akad. Nauk SSSR,134, No.5, 1001–1004 (1961).Google Scholar
  4. 4.
    A. D. Aleksandrov, “Uniqueness conditions and estimates of the solution of the Dirichlet problem,” Vestnik. Leningr. Gos. Univ., No. 13, 5–59 (1963).Google Scholar
  5. 5.
    A. D. Aleksandrov, “Certain estimates of the solutions of the Dirichlet problem,” Vestnik Leningr. Gos. Univ., No. 7, 19–29 (1967).Google Scholar
  6. 6.
    O. A. Ladyzhenskaya and N. N. Ural'tseva, “On the Hölder continuity of solutions and their derivatives for linear and quasilinear equations of elliptic and parabolic types,” Trudy MIAN SSSR, No. 73, 172–220 (1962).Google Scholar
  7. 7.
    S. G. Mikhlin, The Problem of the Minimum of a Quadratic Functional [in Russian], Gostekhizdat, Moscow-Leningrad (1952).Google Scholar
  8. 8.
    M. I. Vishik, “The method of orthogonal and direct expansions in the theory of elliptic differential equations,” Matem. Sb.,25, 189–234 (1949).Google Scholar
  9. 9.
    O. A. Oleinik, “On equations of elliptic and parabolic type with discontinuous coefficients,” Uspekhi Matem. Nauk.,14, No.5, 164–166 (1959).Google Scholar
  10. 10.
    O. A. Oleinik, “Solutions of fundamental boundary value problems for equations of second order with discontinuous coefficients,” Dokl. Akad. Nauk SSSR,124, No.6, 1219–1222 (1959).Google Scholar
  11. 11.
    O. A. Oleinik, “On properties of the solutions of certain boundary value problems for equations of elliptic type,” Matem. Sb.,30, No.3, 695–702 (1952).Google Scholar
  12. 12.
    O. A. Oleinik, “On the Dirichlet problem for equations of elliptic type,” Matem. Sb.,24, No.1, 3–14 (1949).Google Scholar
  13. 13.
    K. O. Friedrichs, “Spektraltheorie halbbeschränkter operatoren und anwendung auf dia spektralzerlegung von differentialoperatoren,” Math. Ann.,109, Nos.4-5, 465–487, 685–713 (1934).Google Scholar
  14. 14.
    C. Miranda, “Alcune osservazioni sulla maggiorazione in Lν delle soluzioni deboli delle equazioni elittiche del secondo ordine,” Ann. di Matematica,61, 151–170 (1963).Google Scholar
  15. 15.
    C. B. Morrey Jr., “Second order elliptic equations in several variables and Hölder continuity,” Math. Zeits.,72, 146–164 (1959).Google Scholar
  16. 16.
    G. Stampacchia, “Problemi al contorno ellitici con dati discontinui datati di soluzion in holderiane,” Ann. Mat. Pure Appl.,51, 1–38 (1960).Google Scholar
  17. 17.
    O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural'tseva, Linear and Quasilinear Equations of Parabolic Type [in Russian], Nauka, Moscow (1967).Google Scholar
  18. 18.
    G. Stampacchia, “Le probléme de Dirichlet pour les équations élliptiques du second order a coefficients discontinus,” Ann. Inst. Fourier Grenoble,15, No.1, 189–258 (1965).Google Scholar
  19. 19.
    G. Stampacchia, “Some limit cases of LP-estimates for solutions of second order elliptic equations,” Comm. Pure Appl. Math.,16, 505–510 (1963).Google Scholar
  20. 20.
    G. Stampacchia, “Régularisation des solutions de problemes aux limites élliptiques a données discontinues,” Inter. Symp. on Linear Spaces, Jerusalem, 399–408 (1960).Google Scholar
  21. 21.
    V. A. Il'in and A. I. Shishmarev, “Method of the potential for the Dirichlet and Neumann problems in the case of equations with discontinuous coefficients,” Sib. Matem. Zh.,2, No.1, 46–58 (1961).Google Scholar
  22. 22.
    V. A. Il'in and A. I. Shishmarev, “On the relation between generalized and classical solutions of the Dirichlet problem,” Izv. Akad. Nauk SSSR, Seriya Matem.,24, No.4, 521–530 (1960).Google Scholar
  23. 23.
    H. O. Cordes, “Über die ersta Randwertaufgabe bei quasilinearen differentialgleichungen zweiter ordnung in mehr als zwei variablen,” Math. Ann.,130, 278–312 (1956).Google Scholar
  24. 24.
    V. P. Il'in, “Properties of certain classes of differentiable functions of several variables defined in n-dimensional regions,” Trudy MIAN SSSR,66, 227–363 (1962).Google Scholar
  25. 25.
    V. P. Il'in, “On imbedding theorems,” Trudy MIAN SSSR,53, 359–386 (1959).Google Scholar
  26. 26.
    A. Kh. Gudiev, “Imbedding theorems for the trace in abstract functions,” Dokl. Akad. Nauk SSSR,147, No.4, 764–767 (1962).Google Scholar
  27. 27.
    A. Kh. Gudiev, “On integrals of potential type and imbedding theorems in spaces with mixed norm,” Diff. Urav.,2, No.1, 83–106 (1966).Google Scholar
  28. 28.
    A. Kh. Gudiev, “On the complete continuity of certain linear operators in spaces with mixed norm,” Izv. Akad. Nauk UzSSR, Ser. Fiz. Mat., No. 5, 18–23 (1965).Google Scholar
  29. 29.
    A. Kh. Gudiev, “On the solvability of the first boundary value problem for an elliptic equation with divergent principal part in W2 1,” Dokl. Akad. Nauk SSSR,178, No.3, 518–521 (1968).Google Scholar
  30. 30.
    A. Bendel and R. Panzone, “The spaces LP with mixed norm,” Duke Math. J.,28, No.3, 301–324 (1961).Google Scholar

Copyright information

© Consultants Bureau, a division of Plenum Publishing Corporation 1970

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  • A. Kh. Gudiev

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