Journal of Mathematical Sciences

, Volume 158, Issue 4, pp 453–604

Equations with nonnegative characteristic form. II

Article

Abstract

This monograph consists of two volumes and is devoted to second-order partial differential equations (mainly, equations with nonnegative characteristic form). A number of problems of qualitative theory (for example, local smoothness and hypoellipticity) are presented.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    A. D. Aleksandrov, “Investigations on the maximum principle. I,” Izv. Vyssh. Uchebn. Zaved. Mat., 5, No. 6, 126–157 (1958).Google Scholar
  2. 2.
    A. D. Aleksandrov, “Investigations on the maximum principle. II,” Izv. Vyssh. Uchebn. Zaved. Mat., 3, No. 10, 3–12 (1959).Google Scholar
  3. 3.
    A. D. Aleksandrov, “Investigations on the maximum principle. III,” Izv. Vyssh. Uchebn. Zaved. Mat., 5, No. 12, 16–32 (1959).Google Scholar
  4. 4.
    A. D. Aleksandrov, “Investigations on the maximum principle. IV,” Izv. Vyssh. Uchebn. Zaved. Mat., 3, No. 16, 3–15 (1960).Google Scholar
  5. 5.
    A. D. Aleksandrov, “Investigations on the maximum principle. V,” Izv. Vyssh. Uchebn. Zaved. Mat., 5, No. 18, 16–26 (1960).Google Scholar
  6. 6.
    P. S. Aleksandrov (Ed.), Hilbert Problems [in Russian], Nauka, Moscow (1969), pp. 216–219.Google Scholar
  7. 7.
    O. Arena, “Problemi parabolici in domini non limitati,” Le Matematiche, 29 (1974).Google Scholar
  8. 8.
    P. G. Aronson, “On the initial value problem for parabolic systems of differential equations,” Bull. Am. Math. Soc., 65, No. 5, 310–318 (1958).MathSciNetGoogle Scholar
  9. 9.
    P. G. Aronson, “Uniqueness of solutions of the initial value problem for parabolic systems of differential equations, ” J. Math. Mech., 11, No. 5, 403–420 (1962).MATHMathSciNetGoogle Scholar
  10. 10.
    K. I. Babenko, “On a new quasianalyticity problem and Fourier transform of entire functions,” Tr. Mosk. Mat. Obshch., 5, 523–542 (1958).MathSciNetGoogle Scholar
  11. 11.
    M. S. Baouendi, “Sur une classe d’opérateurs elliptiques dégénérant au bord,” C. R. Acad. Sci. Paris Sér. A–B, 262, A3337–A340 (1966).MathSciNetGoogle Scholar
  12. 12.
    M. S. Baouendi, “Sur une classe d’opérateurs elliptiques dégénérés,” Bull. Soc. Math. Fr., 95, 45–87 (1967).MATHMathSciNetGoogle Scholar
  13. 13.
    M. S. Baouendi and C. Goulaouic, “Nonanalytic hypoellipticity for some degenerate elliptic operators,” Bull. Am. Math. Soc., 78, No. 3, 483–486 (1972).MATHMathSciNetGoogle Scholar
  14. 14.
    M. S. Baouendi and P. Grisvard, “Sur une équation d’évolution changeant de type,” J. Funct. Anal., 2, 352–367 (1968).MATHMathSciNetGoogle Scholar
  15. 15.
    M. S. Baouendi and P. Grisvard, “Sur une équation d’évolution changeant de type,” C. R. Acad. Sci. Paris Sér. A–B, 265, A556–A558 (1967).MATHMathSciNetGoogle Scholar
  16. 16.
    I. S. Berezin, “On Cauchy’s problem for linear equations of the second order with initial conditions on a parabolic line,” Mat. Sb., 24, No. 66, 301–320 (1949); English transl., Am. Math. Soc. Transl., I, No. 4, 415–439 (1962).MathSciNetGoogle Scholar
  17. 17.
    S. N. Bernšteĭn, “Sur une généralisation des théorèmes de Liouville et de M. Picard,” C. R. Acad. Sci. Paris, 151, 635–638 (1910).Google Scholar
  18. 18.
    S. N. Bernstein, “Sur la nature analytique des solutions des équations aux dériveés partielles des second ordre,” Math. Ann., 59, 20–76 (1904).MATHMathSciNetGoogle Scholar
  19. 19.
    L. Bers, “Mathematical aspects of subsonic and transonic gas dynamics,” Surveys in Appl. Math., Vol. 3, Wiley, New Yoik; Chapman and Hall, London (1958).MATHGoogle Scholar
  20. 20.
    L. Bers, F. John, and M. Schechter, “Partial differential equations,” Lectures in Appl. Math., Vol. 3, Interscience, New York (1964).MATHGoogle Scholar
  21. 21.
    L. Bers, F. John, and M. Shechter, Partial Differential Equations [Russian translation], Mir, Moscow (1966).MATHGoogle Scholar
  22. 22.
    P. Bolley and J. Camus, “Études de la régularité de certains probèmes elliptiques dégénérés dans des ouverts non réguliers, par la méthode de réflexion,” C. R. Acad. Sci. Paris Sér. A–B, 268, A1462–A1464 (1969).MATHMathSciNetGoogle Scholar
  23. 23.
    J.-M. Bony, “Principe du maximum, inégalité de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés,” Ann. Inst. Fourier (Grenoble), 19, fasc. 1, 177–304 (1969).MathSciNetGoogle Scholar
  24. 24.
    J.-M. Bony, “Probéeme de Dirichlet et inégalité de Harnack pour une classe d’opérateurs elliptiques dégénérés du second ordre,” C. R. Acad. Sci. Paris Sér. A–B, 266, A830–A833 (1968).MATHMathSciNetGoogle Scholar
  25. 25.
    J.-M. Bony, “Sur la propagation des maximums et l’unicité du probléme de Cauchy pour les opérateurs elliptiques dégénérés du second ordre,” C. R. Acad. Sci. Paris Sér. A–B, 266, A763–A765 (1968).MATHMathSciNetGoogle Scholar
  26. 26.
    N. I. Chaus, “On the uniqueness of the solution of the Cauchy problem for a differential equation with constant coefficients,” Ukr. Mat. Zh., 16, No. 3, 417–421 (1964).MATHMathSciNetGoogle Scholar
  27. 27.
    N. I. Chaus, “On the uniqueness of the solution of the Cauchy problem for systems of partial differential equations,” Ukr. Mat. Zh., 17, No. 1, 126–130 (1965).MATHMathSciNetGoogle Scholar
  28. 28.
    N. I. Chaus, “Classes of the uniqueness for the solution of the Cauchy problem and the representation of positive-definite kernels,” In: Proc. Semin. on Funct. Anal., Issue 1, Inst. Mat. Akad. Nauk USSR (1968), pp. 170–273.Google Scholar
  29. 29.
    Chi Min-Yu, “The Cauchy problem for a class of hyperbolic equations with initial data on a tins of parabolic degeneracy,” Acta Math. Sin., 8, 521–530 (1958).Google Scholar
  30. 30.
    J. Cohn and L. Nirenberg, “Algebra of pseudodifferential operators,” In: “Pseudodifferential Operators”, A Collection of Translations [Russian translation], Mir, Moscow (1967), pp. 10–62.Google Scholar
  31. 31.
    R. Courant, Methods of Mathematical Physics. Vol. 2: Partial Differential Equations, Interscience, New York (1962).Google Scholar
  32. 32.
    R. Courant, Partial Differential Equations [Russian translation], Mir, Moscow (1964).MATHGoogle Scholar
  33. 33.
    R. Denk and L. R. Volevich, “A new class of parabolic problems connected with Newton’s polygon,” Uch. Zap., Ser. Math. Mech., 1, 146–159 (2005).Google Scholar
  34. 34.
    M. Derridj, “Sur une classe d’opérateurs différentiels hypoelliptiques a cosfficients analytiques,” In: Séminaire Goulaouic–Schwarts 1970/71. Exposé No. 12, École Polytechnique, Centre de Mathématiques, Paris.Google Scholar
  35. 35.
    A. Douglis and L. Nirenberg, “Interior estimates for elliptic systems of partial differential equations,” Commun. Pure Appl. Math., 8, 506–638 (1955).MathSciNetGoogle Scholar
  36. 36.
    W. S. Edelstein, “A spatial decay for the heat equation,” J. Apl. Math. Phys., 20, 900 (1969).MATHMathSciNetGoogle Scholar
  37. 37.
    Yu. V. Egorov, “Hypoelliptic pseudodifferential operators,” Dokl. Akad. Nauk SSSR, 168, 1242–1244 (1966).MathSciNetGoogle Scholar
  38. 38.
    Yu. V. Egorov, “On subelliptic pseudodifferential operators,” Dokl. Akad. Nauk SSSR, 188, 20–22 (1969).MathSciNetGoogle Scholar
  39. 39.
    Yu. V. Egorov, “The canonical transformations of pseudodifferential operators,” Usp. Mat. Nauk, 24, No. 5 (149), 235–236 (1969).MATHGoogle Scholar
  40. 40.
    Yu. V. Egorov, “Pseudodifferential operators of the principal type,” Mat. Sb., 73, No. 3, 356–374 (1967).MathSciNetGoogle Scholar
  41. 41.
    Yu. V. Egorov, “On a certain class of pseudodifferential operators,” Dokl. Akad. Nauk SSSR, 182, No. 6, 1251–1263 (1968).Google Scholar
  42. 42.
    Yu. V. Egorov and V. A. Kondrat’ev, “The oblique derivative problem,” Mat. Sb., 78, No. 120, 148–176 (1969).MathSciNetGoogle Scholar
  43. 43.
    S. D. Eidel’man, “On the Cauchy problem for parabolic systems,” Dokl. Akad. Nauk SSSR, 98, No. 6, 913–915 (1954).MathSciNetGoogle Scholar
  44. 44.
    S. D. Eidel’man, “Estimates of solutions of parabolic systems and some applications,” Mat. Sb., 33, No. 1, 57–72 (1954).MathSciNetGoogle Scholar
  45. 45.
    S. D. Eidel’man and S. D. Ivasishen, “2b-Parabolic systems,” In: Proc. Semin. on Functional Analysis, Issue 1, Inst. Mat. Akad. Nauk USSR (1968), pp. 3–135, 271–273.Google Scholar
  46. 46.
    L. P. Eisenhart, Continuous Groups of Transformations, Princeton Univ. Press, Princeton, N. J. (1933).MATHGoogle Scholar
  47. 47.
    G. M. Fateeva, “The Cauchy problem and boundary-value problem for linear and quasilinear degenerate second-order hyperbolic equations,” Dokl. Akad. Nauk SSSR, 172, 1278–1281 (1967).MathSciNetGoogle Scholar
  48. 48.
    G. M. Fateeva, “Boundary-value problems for degenerate quasilinear parabolic equations,” Mat. Sb., 76, No. 118, 537–565 (1968).MathSciNetGoogle Scholar
  49. 49.
    V. S. Fediĭ, “Estimates in H (S) norms and hipoellipticity,” Dokl. Akad. Nauk SSSR, 193, 301–303 (1970).MathSciNetGoogle Scholar
  50. 50.
    G. Fichera, “Sulle equazioni differenziali lineari elliptico-paraboliche del secondo ordine,” Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Ser. I (8), 5, 1–30 (1956).MathSciNetGoogle Scholar
  51. 51.
    G. Fichera, “On a unified theory of boundary-value problems for elliptic-parabolic equations of second order,” In: Boundary Problems. Differential Equations, Univ. of Wisconsin Press, Madison, Wisconsin (1960), pp. 97–120.Google Scholar
  52. 52.
    J. N. Flavin, “On Knowles version of Saint-Venant’s principle in two-dimensional elastostatics,” Arch. Ration. Mech. Anal., 53, No. 4, 366–375 (1974).MATHMathSciNetGoogle Scholar
  53. 53.
    J. N. Franklin and E. R. Rodemich, “Numerical analysis of an elliptic-parabolic partial differential equation,” SIAM J. Numer. Anal., 5, 680–716 (1968).MathSciNetGoogle Scholar
  54. 54.
    M. I. Freidlin, “Markov processes and differential equations,” In: Theory of Probability. Mathematical Statistics. Theoretical Cybernetics, Akad. Nauk SSSR Inst. Nauch. Inform., Moscow (1967), pp. 7–58.Google Scholar
  55. 55.
    M. I. Freidlin, “The first boundary-value problem for degenerating elliptic differential equations,” Usp. Mat. Nauk, 15, No. 2 (92), 204–206 (1960).Google Scholar
  56. 56.
    M. I. Freidlin, “On the formulation of boundary-value problems for degenerating elliptic equations,” Dokl. Akad. Nauk SSSR, 170, 282–285 (1966).MathSciNetGoogle Scholar
  57. 57.
    M. I. Freidlin, “The stabilization of the solutions of certain parabolic equations and systems,” Mat. Zametki, 3, 85–93 (1968).MATHMathSciNetGoogle Scholar
  58. 58.
    M. I. Freidlin, “Quasilinear parabolic equations and measures on a function space,” Funkts. Anal. Prilozhen., 1, No. 3, 74–82 (1967).MathSciNetGoogle Scholar
  59. 59.
    A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, N. J. (1964).MATHGoogle Scholar
  60. 60.
    K. O. Friedrichs, “Pseudodifferential operators. An introduction,” Courant Lect. Notes Math., New York University (1970).Google Scholar
  61. 61.
    L. Gårding, “Dirichlet’s problem for linear elliptic partial differential equations,” Math. Scand., 1, 55–72 (1953)MATHMathSciNetGoogle Scholar
  62. 62.
    M. Geissert, M. Grec, M. Hieber, and E. V. Radkevich, “The model problem associated to the Stefan problem with surface tension: An approach via Fourier–Laplace multipliers,” In: The Proceedings of a Conference in Cortona, Marcel Dekker (2006).Google Scholar
  63. 63.
    I. M. Gel’fand and G. E. Shilov, Generalized Functions, Vol. 1: Operations on them, Fizmatgiz, Moscow (1958); English transl., Academic Press, New York (1964).Google Scholar
  64. 64.
    I. M. Gel’fand and G. E. Shilov, Generalized Functions, Vol. 2: Spaces of Fundamental Functions, Fizmatgiz, Moscow (1958); English transl., Academic Press, New York (1964).Google Scholar
  65. 65.
    I. M. Gel’fand and G. E. Shilov, “Fourier transform of fast growing functions and the questions of the uniqueness for solutions of the Cauchy problem,” Usp. Mat. Nauk, 8, No. 6, 3–54 (1953).MATHGoogle Scholar
  66. 66.
    S. Gellerstedt, “Sur une équation linéaire aux dérivées partielles de type mixte,” Ark. Mat. Astr. Fys., 25A (1937).Google Scholar
  67. 67.
    T. G. Genčev, “Ultraparabolic equations,” Dokl. Akad. Nauk SSSR, 151, 265–268 (1963).MathSciNetGoogle Scholar
  68. 68.
    V. P. Glushko, “Coerciveness in L 2 of general boundary-value problems for a degenerate second order elliptic equation,” Funkts. Anal. Prilozhen., 2, No. 3, 87–88 (1968).MATHGoogle Scholar
  69. 69.
    M. Yu. Granov and A. S. Shamaev, “Construction and asymptotic analysis of effective assigment of optimal control of investment portfolios,” in press.Google Scholar
  70. 70.
    B. Grec and E. V. Radkevich, “Method of Newton polygon and local solvability of free-boundary problems,” Tr. Sem. I. G. Petrovskogo, in press.Google Scholar
  71. 71.
    V. V. Grushin, “On a class of elliptic pseudodifferential operators degenerate on a submanifold,” Sb. Math., 13, 155–185 (1971).MATHGoogle Scholar
  72. 72.
    B. Hanouzet, “Régularité pour une classe d’opérateurs eliiptiques dégénérés du deuxième ordre,” C. R. Acad. Sci. Paris Sér. A–B, 268,1177-1179 (1969).MATHMathSciNetGoogle Scholar
  73. 73.
    G. Hellwig, “Anfangs- und Randwertprobleme bei partiellen Differentialgleichungen von wechselndem Typus auf den Rändern,” Math. Z., 58, 337–357 (1953).MATHMathSciNetGoogle Scholar
  74. 74.
    D. Hilbert, Grundlagen der Geometrie, 7th ed., Teubner, Leipzig (1930).MATHGoogle Scholar
  75. 75.
    D. Hilbert, “Über die Darstellung definiter Formen als Summen von Formenquadraten,” Math. Ann., 32, 342–350 (1888).MathSciNetGoogle Scholar
  76. 76.
    E. Hille, “The abstract Cauchy problem and Cauchy problem for parabolic differential equations,” J. Anal. Math., 3, 81–196 (1953–1954).MathSciNetGoogle Scholar
  77. 77.
    E. Holmgren, “Sur les solutions quasianalytiques d’l’equations de la chaleur,” Ark. Mat., 18, 64–95 (1924).Google Scholar
  78. 78.
    E. Hopf, “Elementare Bemerkungen über die Lösungen partiellen Differentialgleichungen zweiter Ordnung vom elliptischen Typus,” S.-B. Preuss. Akad. Wiss., 19, 147–152 (1927).Google Scholar
  79. 79.
    L. Hörmander, “On the theory of general partial differential operators,” Acta Math., 94, 161–248 (1955).MATHMathSciNetGoogle Scholar
  80. 80.
    L. Hörmander, “On interior regularity of the solutions of partial differential equations,” Commun. Pure Appl. Math., 11, 197–218 (1958).MATHGoogle Scholar
  81. 81.
    L. Hörmander, “Hypoelliptic differential operators,” Ann. Inst. Fourier (Grenoble), 11, 477–492 (1961).MATHMathSciNetGoogle Scholar
  82. 82.
    L. Hörmander, “Pseudo-differential operators and hypoelliptic equations,” Proc. Sympos. Pure Math., Vol. 10, Amer. Math. Soc, Providence, R. I. (1967) pp. 138–183.Google Scholar
  83. 83.
    L. Hörmander, “Pseudo-differential operators,” Commun. Pure Appl. Math., 18, 501–517 (1965).MATHGoogle Scholar
  84. 84.
    L. Hörmander, “Hypoelliptic second order differential equations,” Acta Math., 119, 147–171 (1967).MATHMathSciNetGoogle Scholar
  85. 85.
    L. Hörmander, Linear Partial Differential Operators, Die Grundlehren der Math. Wissenschaften, Band 116, Academic Press, New York; Springer-Verlag, Berlin (1963).MATHGoogle Scholar
  86. 86.
    L. Hörmander, Linear Partial Differential Operators [Russian translation], Mir, Moscow (1965).MATHGoogle Scholar
  87. 87.
    L. H. Hörmander, “Non-elliptic boundary-value problems,” In: “Pseudifferential Operators,” A Collection of Translations [Russian translation], Mir, Moscow (1967), pp. 166–296.Google Scholar
  88. 88.
    A. M. Il’in, “On a class of ultraparabolic equations,” Dokl. Akad. Nauk SSSR, 159, 1214–1217 (1964).MathSciNetGoogle Scholar
  89. 89.
    A. M. Il’in, “On Dirichlet’s problem for an equation of elliptic type degenerating on some set of interior points of a region,” Dokl. Akad. Nauk SSSR, 102, 9–12 (1955).MATHMathSciNetGoogle Scholar
  90. 90.
    A. M. Il’in, “Degenerate elliptic and parabolic equations,” Mat. Sb., 50, No. 92, 443–498 (1960).MathSciNetGoogle Scholar
  91. 91.
    A. M. Il’in, “Degenerating elliptic and parabolic equations,” Nauchn. Dokl. Vyssh. Shkoly. Fiz.-Mat. Nauki, 2, 48–54 (1958).Google Scholar
  92. 92.
    A. M. Il’in, A. S. Kalashnikov, and O. A. Oleinik, “Second-order linear equations of the parabolic type,” Usp. Mat. Nauk, 17, No. 3 (105), 3–146 (1962).Google Scholar
  93. 93.
    S. D. Ivasishen, “The Green matrix for a general nonhomogeneous parabolic problem with boundary conditions of any order,” Dokl. Akad. Nauk SSSR, 206, No. 4, 796–7911 (1972).MathSciNetGoogle Scholar
  94. 94.
    S. D. Ivasishen and S. D. Eidel’man, “Study of the Green matrices for the homogeneous parabolic problem,” Tr. Mosk. Mat. Obshch., 23, 179–234 (1970).MATHGoogle Scholar
  95. 95.
    S. D. Ivasishen and V. P. Lavrenchuk, “On the solvability of the Cauchy problem and some boundary value problems for general parabolic systems in the class of growing functions,” Dokl. Akad. Nauk USSR, 4, 299–303 (1967).Google Scholar
  96. 96.
    S. D. Ivasishen and V. P. Lavrenchuk, “On the correct solvability of general boundary value problems for parabolic systems with growing coefficients,” Ukr. Mat. Zh., 30, No. 1 (1978).Google Scholar
  97. 97.
    V. Ja. Ivriĭ, “The Cauchy problem for nonstrictly hyperbolic equations,” Dokl. Akad. Nauk SSSR, 197, 517–519 (1971).MathSciNetGoogle Scholar
  98. 98.
    T. Jamanaka, “A refinement of the uniqueness bound of solutions of the Cauchy problem,” Funkcial. Ekvac., 11, 75–86 (1968).MathSciNetGoogle Scholar
  99. 99.
    M. V. Keldysh, “On certain cases of degeneration of equations of elliptic type on the boundary of a domain,” Dokl. Akad. Nauk SSSR, 77, 181–183 (1951).Google Scholar
  100. 100.
    J. K. Knowles, “On Saint-Venant’s principle in the two-dimensional linear theory of elasticity,” Arch. Ration. Mech. Anal., 21, No. 1, 1–22 (1966).MathSciNetGoogle Scholar
  101. 101.
    J. K. Knowles, “A Saint-Venant principle for a classe of second-order elliptic boundary value problems,” J. Appl. Math. Phys., 18, No. 4, 473–490 (1967).MATHGoogle Scholar
  102. 102.
    J. K. Knowles, “On the spatial decay of solutions of the heat equation,” J. Appl. Math. Phys., 22, No. 6, 1050–1056 (1971).MathSciNetGoogle Scholar
  103. 103.
    J. J. Kohn, “Pseudo-differential operators and non-elliptic problems,” In: Pseudo-Differential Operators (C. I. M. E., Streza, 1968), Edizioni Cremonese, Rome (1969), pp. 157–165.Google Scholar
  104. 104.
    J. J. Kohn and L. Nirenberg, “Degenerate elliptic-parabolic equations of second order,” Commun. Pure Appl. Math., 20, 797–872 (1967).MATHMathSciNetGoogle Scholar
  105. 105.
    J. J. Kohn and L. Nirenberg, “An algebra of pseudodifferential operators,” Commun. Pure Appl. Math., 18, 269–305 (1965).MATHMathSciNetGoogle Scholar
  106. 106.
    J. J. Kohn and L. Nirenberg, “Non-coercive boundary value problems,” Commun. Pure Appl. Math., 18, 443–492 (1965).MATHMathSciNetGoogle Scholar
  107. 107.
    A. N. Kolmogorov, “Zufällige Bewegungen,” Ann. Math., 35, 116–117 (1934).MathSciNetGoogle Scholar
  108. 108.
    V. A. Kondrat’ev, “Boundary value problems for parabolic equations in closed regions,”Tr. Mosk. Mat. Obs., 15, 400–451 (1966).MATHGoogle Scholar
  109. 109.
    S. N. Kruzhkov, “Boundary value problems for second order elliptic equations,” Mat. Sb., 77, No. 119, 299–334 (1968).MathSciNetGoogle Scholar
  110. 110.
    L. D. Kudryavtsev, “On the solution by the variational method of elliptic equations which degenerate on the boundary of the region,” Dokl. Akad. Nauk SSSR, 108, 16–19 (1956).MATHMathSciNetGoogle Scholar
  111. 111.
    L. D. Kudryavtsev, “Direct and inverse imbedding theorems. Applications to the solution of elliptic equations by variational methods,” Tr. Mat. Inst. Steklova, 55 (1959).Google Scholar
  112. 112.
    O. A. Ladyzhenskaya, “On the uniquenes of the solution of the Cauchy problem for a linear parabolic equation,” Mat. Sb., 27 (69), 175–184 (1950).Google Scholar
  113. 113.
    V. P. Lavrenchuk, “General boundary value problems for parabolic systems with growing coefficients,” Dokl. Akad. Nauk USSR. Ser. A, 3, 238–242 (1968).Google Scholar
  114. 114.
    A. Lax, “On Cauchy’s problem for partial differential equations with multiple characteristics,” Commun. Pure Appl. Math., 9, 135–169 (1956).MATHMathSciNetGoogle Scholar
  115. 115.
    P. D. Lax, “Asymptotic solutions of oscillatory initial value problems,” Duke Math. J., 24, 627–546 (1957).MATHMathSciNetGoogle Scholar
  116. 116.
    E. E. Levi, “Opere,” In: A Cura Dell’unione Matematica Italiana e Col Contributo del Consiglio Nazional delle Ricerche, 2 vols., Edizioni Cremonese, Rome (1959), (1960), pp. 15–18.Google Scholar
  117. 117.
    E. E. Levi, “Sull’equazione del calore,” Ann. Mat. Pure Appl. Ser. 3, 14, 187–264 (1908).Google Scholar
  118. 118.
    H. Lewy, “Neuer Beweis des analytischen Charakters der Lösungen elliptischer Differentialgleichungen,” Math. Ann., 102, 609–619 (1929).MathSciNetGoogle Scholar
  119. 119.
    J. L. Lions, Quelques methodes de résolution des problèmes aux limites non linéaires, Dunod; Gauthier-Villars, Paris (1969).MATHGoogle Scholar
  120. 120.
    V. E. Lyantse, “On the Cauchy problem in the field of functions of the real variable,” Usp. Mat. Nauk, 1, No. 4, 42–63 (1949).MATHGoogle Scholar
  121. 121.
    B. Malgrange, “Sur une classe d’opérateurs différentiels hypoelliptiques,” Bull. Soc. Math. Fr., 85, 283–306 (1957).MATHMathSciNetGoogle Scholar
  122. 122.
    M. L. Marinov, “A priori estimates for solutions of boundary value problems for general parabolic systems in unbounded domains,” Usp. Mat. Nauk, 32, No. 2, 217–218 (1977).MATHMathSciNetGoogle Scholar
  123. 123.
    M. L. Marinov, “The existence of solutions of the boundary value problem for general parabolic systems in an unbounded domain,” Vestn. MGU. Ser. Mat., Mekh., No. 6, 56–63 (1977).Google Scholar
  124. 124.
    V. G. Maz’ya, “The degenerate problem with oblique derivative,” Usp. Mat. Nauk, 25, No. 2 (152), 275–276 (1970).MATHGoogle Scholar
  125. 125.
    V. G. Maz’ya and B. P. Panejah, “Degenerate elliptic pseudodifferential operators on a smooth manifold without boundary,” Funkts. Anal. Prilozhen., 3, No. 2, 91–92 (1969).Google Scholar
  126. 126.
    V. P. Mikhajlov, “An existence and uniqueness theorem for the solution of a certain boundary value problem for a parabolic equation in a domain with singular points on the boundary,” Tr. Mat. Inst. Steklova, 91, 47–58 (1967).Google Scholar
  127. 127.
    S. G. Mikhlin, “Degenerate elliptic equations,” Vestn. LGU, 9, No. 8, 19–48 (1954).Google Scholar
  128. 128.
    S. G. Mikhlin, The Minimum Problem of a Quadratic Functional, Holden-Day, Inc., San Francisco–London–Amsterdam (1965).MATHGoogle Scholar
  129. 129.
    C. Miranda, “Equazioni alle derivate parziali di ttpo ellittico,” In: Ergebnisse der Mathematik und ihrer Grenzgebiete, Heft 2, Springer-Verlag, Berlin (1955).Google Scholar
  130. 130.
    S. Mizohata, “Solutions nulles et solutions non analytiques,” J. Math. Kyoto Univ., 1, No. 1, 271–302 (1962).MATHMathSciNetGoogle Scholar
  131. 131.
    S. Mizohata and Y. Ohya, “Sur la condition de E. E. Levi concernant des équations hyperboliques,” Publ. Res. Inst. Math. Sci. Ser. A, 4, 511–526 (1968).MATHMathSciNetGoogle Scholar
  132. 132.
    C. Morrey and L. Nirenberg, “On the analyticity of the solutions of linear elliptic systems of partial differential equations,” Commun. Pure Appl. Math., 10, 271–290 (1957).MATHMathSciNetGoogle Scholar
  133. 133.
    T. Nagano, “Linear differential systems with singularities and an application to transitive Lie algebras,” J. Math. Soc. Jpn., 18, 398–404 (1966).MATHMathSciNetGoogle Scholar
  134. 134.
    A. B. Nersesyan, “The Cauchy problem for a second-order hyperbolic equation degenerating on the initial hyperplane,” Dokl. Akad. Nauk SSSR, 181, 798–801 (1968).MathSciNetGoogle Scholar
  135. 135.
    S. M. Nikol’skii, Approximation of functions of several variables, and imbedding theorems [in Russian], Nauka, Moscow (1969).MATHGoogle Scholar
  136. 136.
    L. Nirenberg, “A strong maximum principle for parabolic equations,” Commun. Pure Appl. Math., 6, 167–177 (1953).MATHMathSciNetGoogle Scholar
  137. 137.
    O. A. Oleinik, “Alcuni risultati sulle equazioni lineari e quasi lineari ellittico-paraboliche a derivate parziali del secondo ordine,” Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Nat. Ser. I (8), 40, 775–784 (1966).MathSciNetGoogle Scholar
  138. 138.
    O. A. Oleinik, “A boundary value problems for elliptic-parabolic linear equations,” Lecture Ser., No. 46, University of Maryland Inst. Fluid Dynamics and Appl. Math. (1965).Google Scholar
  139. 139.
    O. A. Oleinik, “The Cauchy problem and the boundary value problem for second-order hyperbolic equations degenerating in a domain and on its boundary,” Dokl. Akad. Nauk SSSR, 169, 525–528 (1966).MathSciNetGoogle Scholar
  140. 140.
    O. A. Oleinik, “On second order hyperbolic equations degenerating in the interior of a region and on its boundary,” Usp. Mat. Nauk, 24, No. 2 (146), 229–230 (1969).MathSciNetGoogle Scholar
  141. 141.
    O. A. Oleinik, “Mathematical problems of boundary layer theory,” Usp. Mat. Nauk, 23, No. 3 (141), 3–65 (1968).MathSciNetGoogle Scholar
  142. 142.
    O. A. Oleinik, “A problem of Fichera,” Dokl. Akad. Nauk SSSR, 157, 1297–1300 (1964).MathSciNetGoogle Scholar
  143. 143.
    O. A. Oleinik, “On the smoothness of solutions of degenerate elliptic and parabolic equations,” Dokl. Akad. Nauk SSSR, 163, 577–580 (1965).MathSciNetGoogle Scholar
  144. 144.
    O. A. Oleinik, “Linear equations of second order with nonnegative characteristic form,” Mat. Sb., 69, No. 111, 111–140 (1966).MathSciNetGoogle Scholar
  145. 145.
    O. A. Oleinik, “On the equations of elliptic type degenerating on the boundary of a region,” Dokl. Akad. Nauk SSSR, 87, 885–888 (1952).MathSciNetGoogle Scholar
  146. 146.
    O. A. Oleinik, “On properties of solutions of certain boundary problems for equations of elliptic type,” Mat. Sb., 30, No. 72, 695–702 (1952).MathSciNetGoogle Scholar
  147. 147.
    O. A. Oleinik, “Discontinuous solutions of non-linear differential equations,” Usp. Mat. Nauk, 12, No. 3 (75), 3–73 (1957).MathSciNetGoogle Scholar
  148. 148.
    O. A. Oleinik, “On the equations of unsteady filtration,” Dokl. Akad. Nauk SSSR, 113, 1210–1213 (1957).MathSciNetGoogle Scholar
  149. 149.
    O. A. Oleinik, “On the Cauchy problem for weakly hyperbolic eguations,” Commun. Pure Appl. Math., 23, 569–586 (1970).MathSciNetGoogle Scholar
  150. 150.
    O. A. Oleinik, “On the uniqueness of solutions of the Cauchy problem for general parabolic systems in the classes of fast growing functions,” Usp. Mat. Nauk, 29, No. 5, 229–230 (1974).MathSciNetGoogle Scholar
  151. 151.
    O. A. Oleinik, “On the uniqueness of solutions of boundary value problems and the Cauchy problem for general parabolic systems,” Dokl. Akad. Nauk SSSR, 220, No. 6, 34–37 (1975).Google Scholar
  152. 152.
    O. A. Oleinik, “On the behavior of solutions of linear parabolic systems of differential equations in unbounded domains,” Usp. Mat. Nauk, 30, No. 2, 219–220 (1975).MathSciNetGoogle Scholar
  153. 153.
    O. A. Oleinik, “On the behaviour of solutions of the Cauchy problem and the boundary value problem for parabolic systems of partial differential equations in unbounded domains,” Rend. Mat. Ser. VI, 8, No. 2, 545–561 (1975).MATHMathSciNetGoogle Scholar
  154. 154.
    O. A. Oleinik, “Analyticity of solutions and related methods of the study of partial differential equations,” Univ. Ann. Appl. Math., 11, No. 2, 151–165 (1975).MathSciNetGoogle Scholar
  155. 155.
    O. A. Oleinik, Lectures on Partial Differential Equations I [in Russian], Moscow State Univ., Moscow (1976).Google Scholar
  156. 156.
    O. A. Oleinik and G. A. Iosif’yan, “On the Saint-Venant principle in the planar elasticity theory,” Dokl. Akad. Nauk SSSR, 239, No. 3, 530–533 (1978).MathSciNetGoogle Scholar
  157. 157.
    O. A. Oleinik and G. A. Iosif’yan, “The Saint-Venant principle for the mixed problem of the elasticity theory and applications,” Dokl. Akad. Nauk SSSR, 233, No. 5, 824–827 (1977).MathSciNetGoogle Scholar
  158. 158.
    O. A. Oleinik and G. A. Iosif’yan, “The Saint-Venant principle in the planar elasticity theory and boundary value problems for the biharmonic equation in unbounded domains,” Sib. Mat. Zh., 19, No. 5, 1154–1165 (1978).MathSciNetGoogle Scholar
  159. 159.
    O. A. Oleinik and G. A. Iosif’yan, “A priori estimates for the first boundary value problem for the system of equations of the elasticity theory and their applications,” Usp. Mat. Nauk, 32, No. 5, 197–198 (1977).Google Scholar
  160. 160.
    O. A. Oleinik, G. A. Iosif’yan, and I. N. Tavkhelidze, “Estimates of the solution of the biharmonic equation in a neighborhood of irregular points of the boundary and at the infinity,” Usp. Mat. Nauk, 33, No. 3, 181–182 (1978).Google Scholar
  161. 161.
    O. A. Oleinik and G. A. Iosif’yan, “On singularities at the boundary points and uniqueness theorems for solutions of the first boundary value problem of elasticity,” Commun. Partial Differ. Equat., 2, No. 9, 937–969 (1977).MATHGoogle Scholar
  162. 162.
    O. A. Oleinik and G. A. Iosif’yan, “Boundary value problems for second order elliptic equations in unbounded domains and Saint-Venant principle,” Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 4, No. 2, 269–290 (1977).MathSciNetGoogle Scholar
  163. 163.
    O. A. Oleinik and G. A. Iosif’yan, “Energy estimates for generalized boundary value problems for second order elliptic equations and applications,” Dokl. Akad. Nauk SSSR, 232, No. 6, 1257–1260 (1977).MathSciNetGoogle Scholar
  164. 164.
    O. A. Oleinik and G. A. Iosif’yan, “An analog of the Saint-Venant principle and the uniqueness of solutions of boundary value problems in unbounded domains for parabolic equations,” Usp. Mat. Nauk, 31, No. 6, 142–166 (1976).MathSciNetGoogle Scholar
  165. 165.
    O. A. Oleinik and N. O. Maksimova, “On the behavior of solutions of nonhomogeneous elliptic systems in unbounded domains,” Tr. Sem. I. G. Petrovskii, 3, 117–137 (1977).MathSciNetGoogle Scholar
  166. 166.
    O. A. Oleinik and T. D. Ventcel’, “The first boundary problem and the Cauchy problem for quasi-linear equations of parabolic type,” Mat. Sb., 41, No. 83, 105–128 (1957).MathSciNetGoogle Scholar
  167. 167.
    O. A. Oleinik and E. V. Radkevich, “On the local smoothness of weak solutions and hypoellipticity of differential equations of second order,” Usp. Mat. Nauk, 26, No. 2 (158), 265–281 (1971).Google Scholar
  168. 168.
    O. A. Oleinik and E. V. Radkevich, “On the analyticity of solutions of linear partial differential equations,” Mat. Sb., 90(132), 592–606 (1973).Google Scholar
  169. 169.
    O. A. Olejnik and E. V. Radkevich, “Analyticity of solutions of linear differential equations and systems,” Sov. Math., Dokl., 13, 1614–1618 (1972).Google Scholar
  170. 170.
    O. A. Olejnik and E. V. Radkevich, “On systems of linear differential equations that have nonanalytic solutions,” Usp. Mat. Nauk, 28, No. 5 (173), 247–248 (1972).Google Scholar
  171. 171.
    O. A. Oleinik and E. V. Radkevich, Analyticity and Theorems on the Behavior of Sulutions of General Ellyptic Systems of Differential Equations in Unbounded Domains [in Russian], Institute for Problems of Mechanics, Russian Academy of Sciences, Preprint No. 47, Moscow (1974).Google Scholar
  172. 172.
    A. A. Oleinik and E. V. Radkevich, “Method of introducing a parameter for studying evelutionary equations,” Usp. Mat. Nauk, 33, No. 5 (203), 7–76 (1978).MathSciNetGoogle Scholar
  173. 173.
    O. A. Oleinik and E. V. Radkevich, “The analyticity and theorems of the Liouville and Phragmen–Lindelöf type for general parabolic systems of differential equations,” Funct. Anal., 8, No. 4, 59–70 (1974).Google Scholar
  174. 174.
    O. A. Oleinik and E. V. Radkevich, “Second-order equations with nonnegative characteristic form,” In: Progress in Science and Tehnology, Series on Contemporary Problems in Mathematics [in Russian], Vol. 18, All-Union Institute for Scientific and Technical Information (VINITI), Akad. Nauk SSSR, Moscow (1971).Google Scholar
  175. 175.
    I. G. Petrovskii, Lectures on Partial Differential Equations [in Russian], 3rd aug. ed., Fizmatgiz, Moscow (1961).Google Scholar
  176. 176.
    I. G. Petrovskii, Lectures on the Theory of Ordinary Differential Equations [in Russian], 5th ed., Nauka, Moscow (1964).Google Scholar
  177. 177.
    R. S. Phillips and L. Sarason, “Elliptic-parabolic equations of the second order,” J. Math. Mech., 17, 891–917 (1967/68).MathSciNetGoogle Scholar
  178. 178.
    R. S. Phillips and L. Sarason, “Singular symmetric positive first order differential operators,” J. Math. Mech., 15, 235–271 (1966).MATHMathSciNetGoogle Scholar
  179. 179.
    M. Picone, “Some forgotten almost sixty years old Lincean notes on the theory of second order linear partial differential equations of the elliptic-parabolic type,” Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Natur. (1968).Google Scholar
  180. 180.
    M. Picone, “Teoremi di unicità nei problemi dei valori al contorno per le equazioni ellittiche e paraboliche,” Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Natur., 22, No. 2, 275–282 (1913).MathSciNetGoogle Scholar
  181. 181.
    M. H. Protter, “The Cauchy problem for a hyperbolic second order equation with data on the parabolic line,” Can. J. Math., 6, 542–553 (1954).MATHMathSciNetGoogle Scholar
  182. 182.
    M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Prentice- Hall, Englewood Cliffs, N. J. (1967).Google Scholar
  183. 183.
    C. Pucci, “Proprietà di massimo e minimo delle soluzioni di equazioni a derivate parziali del secondo ordine di tipo ellittico e parabolico. I,” Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Natur. (8), 23, 370–375 (1957).Google Scholar
  184. 184.
    C. Pucci, “Proprietà di massimo e minimo delle soluzioni di equazioni a derivate parziali del secondo ordine di tipo ellittico e parabolico. II,” Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Natur. (8), 24, 3–6 (1958).Google Scholar
  185. 185.
    E. V. Radkevich, “The second boundary value problem for a second order equation with nonnegative characteristic form,” Vestn. Moskov. Univ. Ser. I Mat. Mekh., 22, No. 4, 3–11 (1967).MATHGoogle Scholar
  186. 186.
    E. V. Radkevich, “A Schauder type estimate for a certain class of pseudo-differential operators,” Usp. Mat. Nauk, 24, No. 1 (145), 199–200 (1969).Google Scholar
  187. 187.
    E. V. Radkevich, “On a theorem of L. Hörmander,” Usp. Mat. Nauk, 24, No. 2 (146), 233–234 (1969).Google Scholar
  188. 188.
    E. V. Radkevich, “A priori estimates and hypoelliptic operators with multiple characteristics,” Dokl. Akad. Nauk SSSR, 187, 274–277 (1969).MathSciNetGoogle Scholar
  189. 189.
    E. V. Radkevich, “Hypoelliptic operators with multiple characteristics,” Mat. Sb., 79, No. 121, 193–216 (1969).MathSciNetGoogle Scholar
  190. 190.
    E. V. Radkevich, “Equations with nonnegative characteristic form. I,” In: Contemporary Mathematics and Its Applications [in Russian], Vol. 55 (2008), in press.Google Scholar
  191. 191.
    P. K. Rashevskii, “On the joinability of any two points of a completely nonholonomic space by an admissible line,” Uch. Zap. Mosk. Gos. Ped. Inst. im. Libknehta Ser. Fiz.-Mat., 2, 83–94 (1938).Google Scholar
  192. 192.
    V. M. Petkov, “Necessary conditions for correctness of the Cauchy problem for hyperbolic systems with multiple characteristics,” Usp. Mat. Nauk, 27, No. 4 (166), 221–222 (1972).MATHMathSciNetGoogle Scholar
  193. 193.
    V. M. Petkov, “Necessary conditions for correctness of the Cauchy problem for non-strictly hyperbolic equations,” Dokl. Akad. Nauk SSSR, 206, 287–290 (1972).MathSciNetGoogle Scholar
  194. 194.
    I. G. Petrovskii, “On some problems of the theory of parial differential equations,” Usp. Mat. Nauk, 1, No. 3-4, 44–70 (1946).MathSciNetGoogle Scholar
  195. 195.
    I. G. Petrovskii, “On the Cauchy problem for systems of linear partial differential equations in the domain of non-analytic functions,” Bull. MGU, Sect. A, 1, No. 7 (1938).Google Scholar
  196. 196.
    I. G. Petrowsky, “Sur l’analyticité des solutions des systems d’équations différentielles,” Mat. Sb., 5(47), 3–70 (1939).MATHMathSciNetGoogle Scholar
  197. 197.
    P. I. Plotnikov, E. V. Ruban, and J. Sokolovski, “Inhomogeneous boundary pproblems for compressible Navier–Stokes equatyions,” in press.Google Scholar
  198. 198.
    P. I. Plotnikov, E. V. Ruban, and J. Sokolovski, “Inhomogeneous boundary pproblems for compressible Navier–Stokes and transprt equatyions,” in press.Google Scholar
  199. 199.
    F. Riesz and B. Sz.-Nagy, “Leçons d’analyse fonctionnelle,” Akad. Kiadó, Budapest (1953).Google Scholar
  200. 200.
    V. S. Ryzhii, “On the uniqueness of the solution for the Cauchy problem for systems parabolic in the sense of I. G. Petrovskii with growing coefficients,” Zap. Mekh.-Mat. Fac. KhGU and Khar’kov Mat. Obshch., 29, Ser. 4 (1963).Google Scholar
  201. 201.
    A. J. C. Barre de Saint-Venant, “De la torsion des prismes,” Mem. Divers Savants, Acad. Sci. Paris, 14, 233–560 (1855).Google Scholar
  202. 202.
    J. Schauder, “Über lineare elliptische Differentialgleichungen zweiter Ordnung,” Math. Z., 38, 257–282 (1934).MathSciNetGoogle Scholar
  203. 203.
    M. Schechter, “On the Dirichlet problem for second order elliptic equations with coefficients singular at the boundary,” Commun. Pure Appl. Math., 13, 321–328 (1960).MATHMathSciNetGoogle Scholar
  204. 204.
    L. Schwartz, Théorie des Distributions. Tomes I, II, Actualite’s Sci. Indust., Nos. 1091, 1122, Hermann, Paris (1950), (1951).Google Scholar
  205. 205.
    L. Schwartz, Methodes Mathematiques pour les Sciences Physiques, Hermann, Paris (1961).MATHGoogle Scholar
  206. 206.
    L. Schwartz, “Les équations d’evolution lieés au produit de composition,” Ann. Inst. Fourier (Grenoble), 2, 19–49 (1950).MATHMathSciNetGoogle Scholar
  207. 207.
    G. E. Shilov, “Local properties of solutions of partial differential equations with constant coefficients,” Usp. Mat. Nauk, 14, No. 5 (89), 3–44 (1959).MATHGoogle Scholar
  208. 208.
    V. G. Sigillito, “On the spatial decay of solutions of parabolic equations,” J. Apl. Math. Phys., 21, 1078 (1970).MATHMathSciNetGoogle Scholar
  209. 209.
    M. M. Smirnov, Degenerating Elliptic and Hyperbolic Equations [in Russian], Nauka, Moscow (1966).MATHGoogle Scholar
  210. 210.
    G. N. Smirnova, “Linear parabolic equations which degenerate on the boundary of the region,” Sib. Mat. Z., 4, 343–358 (1963).MATHMathSciNetGoogle Scholar
  211. 211.
    G. N. Smirnova, “On the classes of uniqueness of the solutions of the Cauchy problem for parabolic equations,” Dokl. Akad. Nauk SSSR, 153, No. 6, 1269–1272 (1963).MathSciNetGoogle Scholar
  212. 212.
    G. N. Smirnova, “The Cauchy problem for parabolic equations degenerating at infinity,” Mat. Sb., 70, No. 4, 591–604 (1966).MathSciNetGoogle Scholar
  213. 213.
    S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics [in Russian], LGU, Leningrad (1950).Google Scholar
  214. 214.
    S. L. Sobolev, “Méthode nouvelle à résoudre le problème de Cauchy pour les équations linéaires hyperboliques normales,” Mat. Sb., 1, No. 43, 39–72 (1936).MATHGoogle Scholar
  215. 215.
    V. A. Solonnikov, “On boundary value problems for linear parabolic differential equations of the general form,” Tr. Mat. Inst. Akad. Nauk SSSR, 83 (1965).Google Scholar
  216. 216.
    V. A. Solonnikov, “On the Green matrices for parabolic boundary value problems,” Zap. Nauch. Semin. LOMI Akad. Nauk SSSR, 14, 256–267 (1969).MATHMathSciNetGoogle Scholar
  217. 217.
    I. M. Sonin, “On the classes of uniqueness for degenerating parabolic equations,” Mat. Sb., 85, No. 4, 459–473 (1971).MathSciNetGoogle Scholar
  218. 218.
    G. Strang and H. Flaschka, “The correctness of the Cauchy problem,” Adv. Math., 6, 347–379 (1971).MATHMathSciNetGoogle Scholar
  219. 219.
    K. Suzuki, “The first boundary value problem and the first eigenvalue problem for the elliptic equations degenerate on the boundary,” Publ. Res. Inst. Math. Sci. Ser. A, 3, 299–335 (1967/68).MathSciNetGoogle Scholar
  220. 220.
    K. Suzuki, “The first boundary value and eigenvalue problems for degenerate elliptic equations. I,” Publ. Res. Inst. Math. Sci. Ser. A, 4, 179–200 (1968/69).MATHMathSciNetGoogle Scholar
  221. 221.
    H. Suzuki, “Analytic-hypoelliptic differential operators of first order in two independent variables,” J. Math. Soc. Jpn., 16, No. 4, 367–374 (1964).MATHCrossRefGoogle Scholar
  222. 222.
    S. Täcklind, “Sur les class quasianalytiques des solutions des equations aux deriveés partielles du type parabolique,” Nova Acta Regial Societatis Schientiarum, Uppsaliensis, Ser. 4, 10, No. 3, 3–55 (1936).Google Scholar
  223. 223.
    A. N. Tikhonov, “Uniqueness theorems for the heat equation,” Mat. Sb., 42, No. 2, 199–215 (1935).MATHGoogle Scholar
  224. 224.
    R. Toupin, “Saint-Venant’s Principle,” Arch. Ration. Mech. Anal., 18, No. 2, 83–96 (1965).MATHMathSciNetGoogle Scholar
  225. 225.
    F. Trèves, “Opérateurs différentiels hypoelliptiques,” Ann. Inst. Fourier (Grenoble), 9, 1–73 (1959).MATHMathSciNetGoogle Scholar
  226. 226.
    Trěves, “Hypoelliptic partial differential equations of principal type with analytic coefficients,” Commun. Pure Appl. Math., 23, No. 4, 637–651 (1970).MATHGoogle Scholar
  227. 227.
    F. Tricomi, “Sulle equazioni lineari alle derivate parziali di secondo ordine, di tipo misto,” Rend. Reale Accad. Lincei. (5), 14, 134–247 (1923).Google Scholar
  228. 228.
    Ya. M. Tsoraev, “On the classes of uniqueness for the solution of the first boundary value problem in unbounded domains and the Cauchy problems for nonuniform parabolic equations,” Vestn. MGU. Ser. Mat., Mekh., No. 3, 38–44 (1970).Google Scholar
  229. 229.
    M. I. Vishik, “On the first boundary problem for elliptic equations degenerating on the boundary of a region,” Dokl. Akad. Nauk SSSR, 93, 9–12 (1953).MATHGoogle Scholar
  230. 230.
    M. I. Vishik, “Boundary-value problems for elliptic equations degenerating on the boundary of a region,” Mat. Sb., 35 (77), 513–568 (1954).Google Scholar
  231. 231.
    M. I. Vishik and V. V. Grushin, “On a class of degenerate elliptic equations,” Mat. Sb., 79 (121), 3–36 (1969).MathSciNetGoogle Scholar
  232. 232.
    M. I. Vishik and V. V. Grushin, “Boundary value problems for elliptic equations degenerate on the boundary of a domain,” Mat. Sb., 80 (122), 455–491 (1969).Google Scholar
  233. 233.
    M. I. Vishik and V. V. Grushin, “Elliptic pseudodifferential operators on a closed manifold which degenerate on a submanifold,” Dokl. Akad. Nauk SSSR, 189, 16–19 (1969).MathSciNetGoogle Scholar
  234. 234.
    M. I. Vishik and G. I. Eskin, “Convolution equations in a bounded region,” Usp. Mat. Nauk, 20, No. 3 (123), 89–152 (1965).Google Scholar
  235. 235.
    L. G. Volevich, “Hypoelliptic equations in convolutions,” Dokl. Akad. Nauk SSSR, 168, 1232–1235 (1966).MathSciNetGoogle Scholar
  236. 236.
    L. F. Volevich, “On a problem of linear programming appearing in differential equations,” Usp. Mat. Nauk, XVIII, No. 3 (111), 155–162 (1963).MATHGoogle Scholar
  237. 237.
    L. R. Volevich, “The Cauchy problem for hypoelliptic differential operators in the classes of fast growing functions,” Usp. Mat. Nauk, 25, No. 1, 191–192 (1970).MATHGoogle Scholar
  238. 238.
    L. R. Volevich and S. G. Gindikin, “The Cauchy problem and related problems for convolution equations,” Usp. Mat. Nauk, 27, No. 4, 65–143 (1872).Google Scholar
  239. 239.
    L. G. Volevich and B. P. Panejah, “Some spaces of generalized functions and embedding theorems,” Usp. Mat. Nauk, 20, No. 1 (121), 3–74 (1965).MATHGoogle Scholar
  240. 240.
    A. I. Vol’pert and S. I. Hudjaev, Cauchy’s problem for second order quasi-linear degenerate parabolic equations,” Mat. Sb., 78 (120), 374–396 (1969).MathSciNetGoogle Scholar
  241. 241.
    N. D. Vvedenskaja, “On a boundary problem for equations of elliptic type degenerating on the boundary of a region,” Dokl. Akad. Nauk SSSR, 91, 711–714 (1953).Google Scholar
  242. 242.
    M. Weber, “The fundamental solution of a degenerate partial differential equation of parabolic type,” Trans. Am. Math. Soc., 71, 24–37 (1951).MATHGoogle Scholar
  243. 243.
    N. Weck, “An explicit Saint-Venant’s principle in three-dimensional elasticity,” Lecture Notes in Math., No. 564, 518–526 (1976).Google Scholar
  244. 244.
    A. Weinstein, “Generalized axially symmetric potential theory,” Bull. Amer. Math. Soc., 59, 20–38 (1953).MATHMathSciNetGoogle Scholar
  245. 245.
    K. Yosida, Functional Analysis, Die Grundlehren der Mathematischen Wissenschaften, Band 123, Academic Press, New York; Springer-Verlag, Berlin (1965).MATHGoogle Scholar
  246. 246.
    E. C. Zachmanoglou, “Propagation of zeros and uniqueness in the Cauchy problem for first order partial differential equations,” Arch. Ration. Mech. Anal., 38, 178–188 (1970).MATHMathSciNetGoogle Scholar
  247. 247.
    Ya. I. Zhitomirskii, “Cauchy problem for parabolic systems of linear partial differential equations with growing coefficients,” Izv. Vuzov. Mat, No. 1 (1959), 55–74.Google Scholar
  248. 248.
    Ya. I. Zhitomirskii, “Classes of the uniqueness for the solution of the Cauchy problem,” Dokl. Akad. Nauk SSSR, 172, No. 6, 1258–1261 (1967).MathSciNetGoogle Scholar
  249. 249.
    Ya. I. Zhitomirskii, “Classes of the uniqueness for the solution of the Cauchy problem for linear equations with fast growing coefficients,” Izv. Akad. Nauk SSSR. Ser. Mat., 31, No. 5, 1159–1178 (1967).MathSciNetGoogle Scholar
  250. 250.
    Ya. I. Zhitomirskii, “Classes of the uniqueness for the solution of the Cauchy problem for linear equations with growing coefficients,” Izv. Akad. Nauk SSSR. Ser. Mat., 31, No. 4, 763–782 (1967).MathSciNetGoogle Scholar
  251. 251.
    G. N. Zolotarev, “On the uniqueness of the solution of the Cauchy problem for systems parabolic in the sense of I. G. Petrovskii,” Izv. Vuzov. Mat., No. 2 (3), 118–135 (1958).Google Scholar
  252. 252.
    G. N. Zolotarev, “On the upper estimates of classes of the uniqueness for the Cauchy problem for systems of partial differential equations,” In: Nauchn. Dokl. Vyssh. Shkoly, No. 2 (1958), pp. 37–40.Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2009

Authors and Affiliations

  1. 1.Moscow State UniversityMoscowRussia

Personalised recommendations