Advertisement

Systems of Linear Differential Equations

  • V. P. Palamodov
Part of the Progress in Mathematics book series (PM, volume 10)

Abstract

The present article is concerned with research in the last five to ten years on systems of linear partial differential equations. The total number of published works in this area, of course, is too great to cover each one in sufficient detail. While consciously refraining from undertaking such a task, I have endeavored to focus a proportionate amount of attention on each facet of the topic insofar as it embodies the characteristics which distinguish the theory of systems from the analogous theory of one equation in one unknown function. For example, considerable space is accorded the -Neumann problem, which is endowed with a specialized character and whose solution has contributed a great deal that is conceptually new to the general theory. On the other hand, the highly-developed theory of boundary-value problems is scarcely touched at all, as its methods pertain by and large to scalar theory.

Keywords

Differential Operator Constant Coefficient Elliptic System Linear Differential Equation Pseudodifferential Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Literature Cited

  1. 1.
    Agranovich, M. S., Elliptic singular integrodifferential operators, Usp. Mat. Nauk, 20 (5): 3–120 (1965).MathSciNetzbMATHGoogle Scholar
  2. 2.
    Agranovich, M. S., Positive boundary-value problems for certain first-order systems, Trudy Moskov. Mat. Obshch., 16: 3–24 (1967).zbMATHGoogle Scholar
  3. 3.
    Agranovich, M. S., On the theory of boundary-value problems for symmetrizable first-order systems, Mat. Sb., 73(2): 161: 197 (1967).Google Scholar
  4. 4.
    Agranovich, M. S. and Dynin, A. S., General boundary-value problems for elliptic systems in a multidimensional domain, Dokl. Akad. Nauk SSSR, 146 (3): 511–514 (1962).Google Scholar
  5. 5.
    Vainberg, B. R. and Grushin, V. V., Uniformly-nonelliptic problems (I), Mat. Sb., 72 (4): 602–636 (1967).Google Scholar
  6. 6.
    Vainberg, B. R. and Grushin, V. V., Uniformly-nonelliptic problems (II), Mat. Sb., 73 (1): 126–154 (1967).Google Scholar
  7. 7.
    Vishik, M. I. and Éskin, G. I., Elliptic equations in convolutions in a bounded domain and their applications, Usp. Mat. Nauk, 22 (1): 15–16 (1967).MathSciNetGoogle Scholar
  8. 8.
    Volevich, L. R., On general systems of differential equations, Dokl. Akad. Nauk SSSR, 132 (1): 20–23 (1960).Google Scholar
  9. 9.
    Volevich, L. R., Regularity of the solutions of systems of differential equations with variable coefficients, Usp. Mat. Nauk, 16 (2): 163–164 (1961).Google Scholar
  10. 10.
    Volevich, L. R., Solvability of boundary-value problems for general elliptic sys- tems, Mat. Sb., 68 (3): 373–416 (1965).Google Scholar
  11. 11.
    Volevich, L. R., Local properties of inhomogeneous pseudodifferential operators, Trudy Moskov. Mat. Obsch., 16: 51–98 (1967).zbMATHGoogle Scholar
  12. 12.
    Vostretsov, B. A., Structure of the analytic solutions of a class of systems of linear partial differential equations with constant coefficients, Dokl. Akad. Nauk SSSR, 161 (6): 1259–1262 (1965).Google Scholar
  13. 13.
    Galakhov, M. A., Nonclassical boundary-value problems for symmetric linear first-order partial differential systems, Differentsial’nye Uravneniya, 1 (12): 1620–1627 (1965).MathSciNetGoogle Scholar
  14. 14.
    Gel’fand, I. M. and Shilov, G. E., Generalized Functions, Part 3: Aspects of the Theory of Differential Equations, Fizmatgiz, Moscow (1958).Google Scholar
  15. 15.
    Gluzberg, E. L, On the Cauchy problem for a countable system of partial differential equations, Proc. First Kazakhstan Sci. Conf. Mathematics and Mechanics, Alma-Ata, 1963, Nauka (1965), pp. 55–57.Google Scholar
  16. 16.
    Golets, B. I. and Verenich, I. I., The Cauchy problem for partial differential systems with variable coefficients, Abstracts of Papers of the Twelfth Scientific Meeting of the Physical and Mathematical Sciences Section of Chernovitsy University, Chernovitsy (1966), pp. 127–129.Google Scholar
  17. 17.
    Golets, B. I. and Eidel’man, S. D., Nine theorems on I. G. Petrovskii-correct systems, Dopovidi Akad. Nauk Ukrain. RSR, No. 9, pp. 1106–1111 (1966).Google Scholar
  18. 18.
    Gorfn, E. A., Solvability of the Cauchy problem, Vest. Moskov. Univ., Mat. Mekh., No. 4, pp. 6–12 (1965).Google Scholar
  19. 19.
    Dezin, A. A., Existence and uniqueness theorems for the solutions of boundary-value problems for partial differential equations in functional spaces, Usp. Mat. Nauk, 16 (3): 21–73 (1959).MathSciNetGoogle Scholar
  20. 20.
    Didenko, V. P., Solvability of certain boundary-value problems for elliptic systems with degenerate order at the boundary, Differentsial’nye Uravneniya, 3 (1): 11–18 (1967).MathSciNetzbMATHGoogle Scholar
  21. 21.
    Zhitarashu, N. V., A priori estimates and the solvability of general boundary-value problems for general elliptic systems with discontinuous coefficients, Dokl. Akad. Nauk SSSR, 165 (1): 24–27 (1965).Google Scholar
  22. 22.
    Zhitarashu, N. V., Schauder estimates and the solvability of general boundary-value problems for general parabolic systems with discontinuous coefficients, Dokl. Akad. Nauk SSSR, 169 (3): 511–514 (1966).Google Scholar
  23. 23.
    Zolotareva, E. V., The Dirichlet problem for some second-order elliptic systems, Differentsial’nye Uravneniya, 3 (1): 59–68 (1967).MathSciNetzbMATHGoogle Scholar
  24. 24.
    Ivasishen, S. D. and Lavrenchuk, V. P., Solvability of the Cauchy problem and certain boundary-value problems for general parabolic systems in a class of increasing functions, Depovidi Akad. Nauk Ukrain. RSR, A, No. 4, pp. 299–303 (1967).Google Scholar
  25. 25.
    Kovach, Yu. I., Application of the theorem of differential inequalities to the Goursat problem for a linear system of partial differential equations, Differentsial’nye Uravneniya, 1 (3): 411–420 (1965).MathSciNetzbMATHGoogle Scholar
  26. 26.
    Kopacek, J. and Suha, M., The Cauchy problem for weakly-hyperbolic systems of linear differential equations with constant coefficients, Casop. Pe`stov. Mat., 91 (4): 431–452 (1966).zbMATHGoogle Scholar
  27. 27.
    Kuz’min, E. M., The Dirichlet problem for elliptic systems in a space, Differentsial’nye Uravneniya, 3 (1): 155–157 (1967).MathSciNetzbMATHGoogle Scholar
  28. Lbdnev, N. A., A new method of solving partial differential equations, Mat. Sb., 22: 205–266 (1948).Google Scholar
  29. 29.
    Lopatinskii, Ya. B., Properties of linear differential operators, Mat. Sb., 17 (2): 267–285 (1945).Google Scholar
  30. 30.
    Lopatinskii, Ya. B., Reduction of a system of differential equations to canonical form, in: Theory and Applications of Mathematics, No. 2, Lvov. Univ., Lvov (1963), pp. 53–64.Google Scholar
  31. 31.
    Matiichuk, M. I. and Éide1’man, S.D., Parabolic systems with coefficients satisfying the Dini condition, Dokl. Akad. Nauk SSSR, 165 (3): 482–485 (1965).Google Scholar
  32. 32.
    Matiichuk, M. I. and Éidel’man, S. D., On the fundamental solutions of elliptic systems, Ukrain. Mat. Zh., 18 (2): 22–41 (1966).Google Scholar
  33. 33.
    Nguen Tkhya Khop, The Dirichlet problem for strongly-connected elliptic systems, Dokl. Akad. Nauk SSSR, 171 (2): 292–295 (1966).Google Scholar
  34. 34.
    Oskolkov, A. M., A priori estimates of the first derivatives for two-dimensional linear strongly-elliptic systems and elliptic mappings, Trudy Mat. Inst. Akad. Nauk SSSR, 92: 182–191 (1966).MathSciNetzbMATHGoogle Scholar
  35. 35.
    Palamodov, V. P., Construction of polynomial ideals and their factor spaces in spaces of infinitely-differentiable functions, Dokl. Akad. Nauk SSSR, 141 (6): 1302–1305 (1961).MathSciNetGoogle Scholar
  36. 36.
    Palamodov, V. P., Systems of differential equations with constant coefficients, Dokl. Akad. Nauk SSSR, 148 (3): 523–526 (1963).MathSciNetGoogle Scholar
  37. 37.
    Palamodov, V. P., Linear Differential Operators with Constant Coefficients, Nauka, Moscow (1967).Google Scholar
  38. 38.
    Palamodov, V. P., Differential operators in the class of convergent power series and the Weierstrass preparation theorem, Funktsional. Analiz i Ego Prilozhen., 2 (3): 58–69 (1968).MathSciNetzbMATHGoogle Scholar
  39. 39.
    Palamodov, V. P., Remark on the exponential representation of the solutions of differential equations with constant coefficients, Mat. Sb., 75 (3): 417–434 (1968).Google Scholar
  40. 40.
    Palamodov, V. P., Differential operators in coherent analytic sheaves, Mat. Sb., 70 (3): 390–422 (1966).Google Scholar
  41. 41.
    Paneyakh, B. P., On general systems of differential equations with constant coefficients, Dokl. Akad. Nauk SSSR, 138 (2): 297–300 (1961).Google Scholar
  42. 42.
    Parasyuk, L. S., Generalized fundamental solution of elliptic systems of differential equations with discontinuous coefficients, Ukrain. Mat. Zh., 18 (4): 124–126 (1966).zbMATHGoogle Scholar
  43. 43.
    Roitberg, Ya. A. and Sheftel’, Z. G., Boundary-value problems with parameter in L for Douglis-Nirenberg-elliptic systems, Ukeain. Mat. Zh., 19 (1): 115–120 (1967).Google Scholar
  44. 44.
    Skripnik, I. V., The a-Neumann problem, Dopovidi Akad. Nauk Ukrain. RSR, No. 3, pp. 295–299 (1966).MathSciNetGoogle Scholar
  45. 45.
    Solomyak, M. Z., On first-order linear elliptic systems, Dokl. Akad. Nauk SSSR, 150 (1): 48–51 (1963).Google Scholar
  46. 46.
    Solonnikov, V. A., General boundary-value problems for systems elliptic in the Douglis-Nirenberg sense (I), Izv. Akad. Nauk SSSR, Ser. Mat., 28 (3): 665–706 (1964).MathSciNetzbMATHGoogle Scholar
  47. 47.
    Solonnikov, V. A., General boundary-value problems for systems elliptic in the Douglis-Nirenberg sense (II), Trudy Mat. Inst. Akad. Nauk SSSR, 92: 233–297 (1966).MathSciNetGoogle Scholar
  48. 48.
    Sochneva, V. A., On the Cauchy problem for systems of partial differential equations with variable coefficients, Differentsial’nye Uravneniya, 2(11):15201530 (1966).Google Scholar
  49. 49.
    Sochneva, V. A., Solutions of general linear systems of partial differential equations analytic on one variable, Izv. Vuzov, Mat., No. 2, pp. 67–73 (1967).Google Scholar
  50. 50.
    Tovmasyan, N. E., Boundary-value problems for second-order elliptic systems of equations that do not satisfy the condition of Ya. B. Lopatinskii, Dokl. Akad. Nauk SSSR, 160 (5): 1028–1031 (1965).Google Scholar
  51. 51.
    Tovmasyan, N. E., General boundary-value problem for second-order elliptic systems with constant coefficients (I), Differentsial’nye Uravneniya, 2 (1): 3–23 (1966).MathSciNetGoogle Scholar
  52. 52.
    Tovmasyan, N. E., General boundary-value problem for second-order elliptic systems with constant coefficients (II), Differentsial’nye Uravneniya, 2 (2): 163–171 (1966).MathSciNetGoogle Scholar
  53. 53.
    Fridlender, V. R., Polynomial matrices and systems of partial differential equations, Izv. Vuzov, Mat., No. 5, pp. 118–123 (1966).Google Scholar
  54. 54.
    Fridlender, V. R. and Khamitov, L. Kh., The generalized Duff-Friedman problem, Izv. Vuzov, Mat., No. 5, pp. 101–107 (1967).Google Scholar
  55. 55.
    Hörmander, L., An Introduction to Complex Analysis in Several Variables, Van Nostrand, Princeton, N. J. (1966).Google Scholar
  56. 56.
    Andreotti, A. and Grauert, H., Finiteness theorems for the cohomologies of complex spaces [in French], Bull. Soc. Math. France, 90: 193–259 (1962).MathSciNetzbMATHGoogle Scholar
  57. 57.
    Andreotti, A. and Vesentini, E., Carleman estimates for the Laplace-Beltrami equations on complex manifolds, Pubis. Math. Inst. Heutes Études Sci., No. 25, pp. 313–362 (1965).zbMATHGoogle Scholar
  58. 58.
    Apostolatos, N. and Kulisch, U., Integrability and integration of overdetermined systems of partial differential equations [in German], Z. Angew. Math. und Mech., 47 (4): 261–268 (1967).MathSciNetzbMATHGoogle Scholar
  59. 59.
    Ash, M. E., The basic estimates of the “5-Neumann problem in the non-Kählerian case, Amer. J. Math., 86 (2): 247–254 (1964).MathSciNetzbMATHGoogle Scholar
  60. 60.
    Atiyah, M. F. and Bott, R., The index problem for manifolds with boundary, in: Differential Analysis, Oxford Univ. Press, London (1964), pp. 175–189.Google Scholar
  61. 61.
    Avantaggiati, A., On the fundamental principal matrices for a class of elliptic and hypoelliptic systems of differential equations [in Italian], Ann. Mat. Pura Appl., 65: 191–237 (1964).MathSciNetzbMATHGoogle Scholar
  62. 62.
    Avantaggiati, A., New contributions to the investigation of the convolution problem for first-order elliptic systems [in Italian], Ann. Mat. Pura Appl., 69: 107–169 (1965).MathSciNetzbMATHGoogle Scholar
  63. 63.
    Brenner, P., The Cauchy problem for symmetric hyperbolic systems in Lp, Math. Scandinay., 19 (1): 27–37 (1966).zbMATHGoogle Scholar
  64. 64.
    Caldwell, W. V., Some relationships between Bers and Beltrami systems and linear elliptic systems of partial differential equations, Canad. J. Math., 17 (4): 627–642 (1965).MathSciNetzbMATHGoogle Scholar
  65. 65.
    Cenkl, B., The elliptic differential operators, Comment. Math. Univ. Carolinae, 8 (2): 175–197 (1967).MathSciNetzbMATHGoogle Scholar
  66. 66.
    Cenkl, B., Vanishing theorem for an elliptic differential operator, J. Differential Geom., 1: 381–418 (1967).MathSciNetzbMATHGoogle Scholar
  67. 67.
    Chen Liang-jin, The Dirichlet problem for a class of systems of degenerate elliptic equations, Chinese Math., 5 (3): 409–417 (1964).Google Scholar
  68. 68.
    Chou Nien-tei, A function theory for a system of first-order elliptic partial differential equations, Sci. Abstr. China, Math. and Phys. Sci., 2 (4): 9–10 (1964).Google Scholar
  69. 69.
    Cinquini, C. M., Systems of partial differential equations in several independent variables [in Italian], Semin. 1962–1963 Analisi, Algebra, Geometria e Topol., Vol. 1, Rome (1965), pp. 101–122.Google Scholar
  70. 70.
    Dombrowski, P., Maximal explicit solutions (Riemann surfaces) of the Cauchy initial-value problem for first-order partial differential equations for one unknown function on C°°-manifolds (I) [in German], Math. Ann., 160 (3): 195–232 (1965).MathSciNetzbMATHGoogle Scholar
  71. 71.
    Dombrowski, P., Maximal explicit solutions (Riemann surfaces) of the Cauchy initial-value problem for first-order partial differential equations for one unknown function on Cs-manifolds (II) [in German], Math. Ann., 160 (4): 257–279 (1965).MathSciNetGoogle Scholar
  72. 72.
    Dombrowski, P., Maximal explicit solutions (Riemann surfaces) of the Cauchy initial-value problem for first-order partial differential equations for one unknown function on C’°-manifolds (III) [in German], Math. Ann., 161 (1): 26–66 (1965).MathSciNetGoogle Scholar
  73. 73.
    Douglis, A. and Nirenberg, L., Interior estimates for elliptic systems of partial equations, Commun. Pure Appl. Math., 8 (4): 503–538 (1955).MathSciNetzbMATHGoogle Scholar
  74. 74.
    Douglis, A., Some applications of the theory of distributions to several complex variables, Seminar on Analytic Functions, Vol. I, Inst. Adv. Study, Princeton (1958), pp. 65–79.Google Scholar
  75. 75.
    Ehrenpreis, A fundamental principle for systems of linear differential equations with constant coefficients and some of its applications, Proc. Internat. Symposium Linear Spaces, Jerusalem, 1960, Acad. Press, Jerusalem; Pergamon, New York (1960).Google Scholar
  76. 76.
    Ehrenpreis, L., Guillemin, V. W. and Sternberg, S., On Spencer’s estimate for 5-Poincaré, Ann. Math., 83 (1): 128–138 (1965).MathSciNetGoogle Scholar
  77. 77.
    Foias, C. and Gussi, G., A uniqueness theorem for the solution of the Cauchy problem for certain linear systems of partial differential equations [in French], Atti Acad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Natur., 29 (6): 509–514 (1960).MathSciNetGoogle Scholar
  78. 78.
    Friedrichs, K. O. and Lax, P. D., Boundary-value problems for first-order operators, Commun. Pure Appl. Math., 18: 355–388 (1965).MathSciNetzbMATHGoogle Scholar
  79. 79.
    Fuglede, B., A priori inequalities connected with systems of partial differential equations, Acta Math., 105 (1961).Google Scholar
  80. 80.
    Garabedian, P. R. and Spencer, D. C., Complex boundary problems, Trans. Amer. Math. Soc., 73: 223–242 (1952).MathSciNetzbMATHGoogle Scholar
  81. 81.
    Garding, L., Energy inequalities for hyperbolic systems, in: Differential Analysis, Oxford Univ. Press, London (1964), pp. 209–225.Google Scholar
  82. 82.
    Garding, L., A variation on the Cauchy majorizing method [in French], Acta Math., 114 (1–2): 143–158 (1965).MathSciNetzbMATHGoogle Scholar
  83. 83.
    Goldschmidt, H., Existence theorems for analytic linear partial differential equations, Ann. Math., 86 (2): 246–270 (1967).MathSciNetzbMATHGoogle Scholar
  84. 84.
    Guillemin, V. W., Some algebraic results concerning the characteristics of over-determined partial differential equations, Amer. J. Math., 90 (1): 270–284 (1968).MathSciNetzbMATHGoogle Scholar
  85. 85.
    Guillemin, V. W. and Sternberg, S., An algebraic model of transitive differential geometry, Bull. Amer. Math. Soc., 70 (1): 16–47 (1964).MathSciNetzbMATHGoogle Scholar
  86. 86.
    Hörmander, L., The Frobenius-Nirenberg theorem, Arkiv Mat., 5 (5): 425–432 (1964).Google Scholar
  87. 87.
    Hörmander, L., L2-estimates and existence theorems for the 5 operator, Acta Math., 113 (1–2): 89–152 (1965).MathSciNetzbMATHGoogle Scholar
  88. 88.
    Hörmander, L., Pseudodifferential operators and nonelliptic boundary problems, Ann. Math., 83 (1): 129–209 (1966).zbMATHGoogle Scholar
  89. 89.
    Hua Loo-Keng and Wu Tze-chien, On the uniqueness theorem of the Dirichlet problem of the linear elliptic system, Sci. Abstr. China. Math. and Phys. Sci., 3 (3): 9–10 (1965).Google Scholar
  90. 90.
    Kodaira, K., On a differential geometric method in the theory of analytic stacks, Proc. Nat. Acad. Sci. USA, 39: 1268–1273 (1953).MathSciNetzbMATHGoogle Scholar
  91. 91.
    Kohn, J. J., Solution of the -(5.-Neumann problem on strongly-pseudoconvex manifolds, Proc. Nat. Acad. Sci. USA, 47: 1198–1202 (1961).MathSciNetzbMATHGoogle Scholar
  92. 92.
    Kohn, J. J., Harmonic integralsonstrongly-pseudoconvex manifolds (I), Ann. Math., 78 (1): 112–148 (1963).MathSciNetzbMATHGoogle Scholar
  93. 93.
    Kohn, J. J., Harmonic integrals on strongly-pseudoconvex manifolds (II), Ann. Math., 79: 450–472 (1964).MathSciNetzbMATHGoogle Scholar
  94. 94.
    Kohn, J. J., Boundaries of complex manifolds, Proc. Conf. Complex Analysis, Minneapolis, 1964, Springer-Verlag, Berlin (1965), pp. 81–94.Google Scholar
  95. 95.
    Kohn, J. J., Differential operators on manifolds with boundary, in: Differential Analysis, Oxford Univ. Press, London (1964), pp. 57–63.Google Scholar
  96. 96.
    Kohn, J. J., Differential complex, Proc. Internat. Congr. Mathematicians, 1966, Mir, Moscow (1968), pp. 402–409.Google Scholar
  97. 97.
    Kohn, J. J. and Nirenberg, L., Noncoercive boundary-value problems, Commun. Pure Appl. Math., 18: 443–492 (1965).MathSciNetzbMATHGoogle Scholar
  98. 98.
    Kohn, J. J., and Rossi, H., On the extension of holomorphic functions from the boundary of a complex manifold, Ann. Math., 81 (3): 451–472 (1965).MathSciNetzbMATHGoogle Scholar
  99. 99.
    Kohn, J. J., and Spencer, D. C., Complex Neumann problems, Ann. Math., 66: 98–140 (1957).MathSciNetGoogle Scholar
  100. 100.
    Komatsu, H., Resolutions by hyperfunctions of sheaves of solutions of differential equations with constant coefficients, Math. Ann., 176 (1): 77–86 (1968).MathSciNetzbMATHGoogle Scholar
  101. 101.
    Kopacek, J., The Cauchy problem for linear hyperbolic systems in Lp, Comment. Math. Univ. Carolinae, 8 (3): 459–462 (1968).MathSciNetGoogle Scholar
  102. 102.
    Kuranishi, M., Sheaves defined by differential equations and application to deformation theory of pseudogroup structures, Amer. J. Math., 86: 379–391 (1964).Google Scholar
  103. 103.
    Kuranishi, M., Lectures on Involutive Systems of Partial Differential Equations (Publ. Soc. Mat. STo Paulo), STo Paulo (1967), 77 pages.Google Scholar
  104. 104.
    Lambert, L., On the Wirtinger system of partial differential equations [in Italian], Rend. Mat. Appl., 23 (3–4): 419–437 (1964).Google Scholar
  105. 105.
    Lax, P. D. and Phillips, R. S., Local boundary conditions for dissipative symmetric linear differential operators, Commun. Pure Appl. Math., 13: 427–455 (1960).MathSciNetzbMATHGoogle Scholar
  106. 106.
    Malgrange, B., Division of distributions, I: Extendable distributions [in French], Sém. Schwartz, Fac. Sci. Paris, 1959–60, 4 année, Paris (1960), 22 [1–21] 5.Google Scholar
  107. 107.
    Malgrange, B., Division of distributions, II: The Lojasiewicz inequality [in French], Sém. Schwartz, Fac. Sci. Paris, 1959–60,4année, Paris (1960), pp. 22 [1–22] 8.Google Scholar
  108. 108.
    Malgrange, B., Division of distributions, III: The principal theorem [in French], Sém. Schwartz, Fac. Sci. Paris, 1959–60, 4 année, Paris (1960), pp. 23 [1–23] 11.Google Scholar
  109. 109.
    Malgrange, B., Division of distributions, IV: Applications [in French], Sém. Schwartz, Fac. Sci. Paris, 1959–60, 4 année, Paris (1960), pp. 25 [1–25] 5.Google Scholar
  110. 110.
    Malgrange, B., On differential systems with constant coefficients [in French], Colloq. Internat. Centre Nat. Rech. Sci., No. 117, pp. 113–122 (1963).Google Scholar
  111. 111.
    Malgrange, B., Differential systems with constant coefficients [in French], Sém. Bourbaki, Secrét. Math., 1962–63, 15(1):246 [01–246]11 (1964).Google Scholar
  112. 112.
    Malgrange, B., Some convexity problems for differential operators with constant coefficients, Sem. Leray, Collège, de France, Paris (1962/63), pp. 190223.Google Scholar
  113. 113.
    Malgrange, B., Some remarks on the notion of convexity for differential operators, in; Differential Analysis, Oxford Univ. Press, London (1964), pp. 163–174.Google Scholar
  114. 114.
    Mansfield, L. E., A Generalization of the Cartan-Kähler Theorem, Doct. Dis-sert., Univ. Washington, 1965, 69 pages; Dissert. Abstr., 26 (8): 4693–4694 (1966).Google Scholar
  115. 115.
    Matzumura, M., Local existence of solutions for certain systems of partial differential equations [in French], Japan J. Math., 32; 13–49 (1962).Google Scholar
  116. 116.
    Matsuura, S., On general systems of partial differential operators with constant coefficients, J. Math. Soc. Japan, 13 (1): 94–103 (1961).MathSciNetzbMATHGoogle Scholar
  117. 117.
    Matsuura, S., A remark on ellipticity of general systems of differential operators with constant coefficients, J. Math. Kyoto Univ., 1 (1): 71–74 (1961).MathSciNetzbMATHGoogle Scholar
  118. 118.
    Milicer-Gruzewska, H., The properties of generalized potentials and a limit problem for elliptic systems [in French], Bull. Acad. Polon. Sci., Sér. Sci. Math. Astron. Phys., 13 (2): 125–133 (1965).MathSciNetzbMATHGoogle Scholar
  119. 119.
    Milicer-Gruzewska, H., On the fundamental solution of elliptic systems with Holder coefficients [in French], Bull. Acad. Polon. Sci., Sér. Sci. Mat. Astron. Phys., 13 (2); 117–124 (1965).MathSciNetzbMATHGoogle Scholar
  120. 120.
    Morrey, C. B., The analytic embedding of abstract real-analytic manifolds, Ann. Math., 68: 159–201 (1958).MathSciNetzbMATHGoogle Scholar
  121. 121.
    Morrey, C. B., The,S-Neumann problem on strongly-pseudoconvex manifolds, in: Differential Analysis, Oxford Univ. Press, London (1964), pp. 81–133.Google Scholar
  122. 122.
    Morrey, C. B. and Nirenberg, L., On the analyticity of the solutions of linear elliptic systems of partial differential equations, Commun. Pure Appl. Math., 10: 271–290 (1957).MathSciNetzbMATHGoogle Scholar
  123. 123.
    Newlander, A. and Nirenberg, L., Complex-analytic coordinates in almost-complex manifolds, Ann., Math., 65: 391–404 (1957).MathSciNetzbMATHGoogle Scholar
  124. 124.
    Nirenberg, L., A complex Frobenius theorem, Sem. Analytic Functions, Vol. 1, Inst. Adv. Study, Princeton (1957), pp. 172–189.Google Scholar
  125. 125.
    Persson, J., A boundary problem for analytic linear systems with data on intersecting hyperplanes, Math. Scandinay., 14 (1): 106–110 (1964).MathSciNetzbMATHGoogle Scholar
  126. 126.
    Phillips, R. S. and Sarason, L., Singular symmetric positive first-order differential operators, J. Math., Mech., 15 (2): 235–271 (1956).MathSciNetGoogle Scholar
  127. 127.
    Schaefer, P. W., On the Cauchy problem for an elliptic system, Arch. Rat. Mech. Anal., 20 (5); 391–412 (1965).MathSciNetzbMATHGoogle Scholar
  128. 128.
    Schechter, M., Systems of partial differential equations in a half-space, Commun. Pure Appt. Math., 17 (4): 423–434 (1964).MathSciNetzbMATHGoogle Scholar
  129. 129.
    Schechter, M., Deformation of structures on manifolds defined by transitive continuous pseudogroups (I), Ann. Math., 76; 306–398 (1962).Google Scholar
  130. 130.
    Schechter, M., Deformation of structures on manifolds defined by transitive continuous pseudogroups (II), Ann. Math., 76: 399–445 (1962).Google Scholar
  131. 131.
    Schechter, M., Harmonic integrals and Neumann problems associated with linear partial differential equations, Outlines of the Joint Soviet-American Symposium on Partial Differential Equations, Novosibirsk (1963), pp. 253–260.Google Scholar
  132. 132.
    Spencer, D. C., Existence of local coordinates for structures defined by elliptic pseudogroups, in: Differential Analysis, Oxford Univ. Press, London (1964), pp. 135–162.Google Scholar
  133. 133.
    Spencer, D. C., De Rham theorems and Neumann decompositions associated with linear partial differential equations, Ann. Inst. Fourier, 14 (1): 1–19 (1964).Google Scholar
  134. 134.
    Spencer, D. C., D.formation of structures on manifolds defined by transitive continuous pseudogroups (II), Ann. Math., 81: 389–450 (1962).Google Scholar
  135. 135.
    Sweeney, W. J., The 6-Poincaré estimate, Pacific J. Math., 20 (3): 559–570 (1967).MathSciNetzbMATHGoogle Scholar
  136. 136.
    Sweeney, W. J., A noncompact Dirichlet norm, Proc. Nat. Acad. Sci. USA, 58 (6): 2193–2195 (1967).zbMATHGoogle Scholar
  137. 137.
    Sweeney, W. J., The D-Neumann problem, Acta Math., 120 (3–4): 223–277 (1968).MathSciNetzbMATHGoogle Scholar
  138. 138.
    Tricomi, F. G., Solution of a problem of Goursat for a particular hyperbolic system of partial differential equations [in Italian], Rend. Ist. Lombardo Sci. Lett. Sci. Mat. Fis. Chim. Geol., 99 (1): 104–109 (1965).MathSciNetzbMATHGoogle Scholar
  139. 139.
    Vaillant, J., Multiple characteristics and bicharacteristics of linear systems of partial differential equations with constant coefficients [in French], Ann. Inst. Fourier, 15 (2): 225–311 (1965).Google Scholar
  140. 140.
    Yamamoto, M., On Cauchy’s problem for a linear system of partial differential equations of first order, Proc. Japan. Acad., 42 (6): 555–559 (1966).zbMATHGoogle Scholar
  141. 141.
    Yamanaka, T., On the Cauchy problem for Kowalevskaja [Kovalevskaya] systems of partial differential equations, Comment. Math. Univ. St. Pauli, 15 (2): 67–89 (1967).zbMATHGoogle Scholar
  142. 142.
    Zerner, M., Singular solutions of partial differential equations [in French], Bull. Soc. Math. France, 91: 302–226 (1963).MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1971

Authors and Affiliations

  • V. P. Palamodov

There are no affiliations available

Personalised recommendations