Skip to main content
SpringerLink
Log in
SpringerLink
Log in

Systems of Linear Differential Equations

  • Chapter
Mathematical Analysis

Part of the book series: Progress in Mathematics ((PM,volume 10))

  • 305 Accesses

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 EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbWexLMBbXgBd9gzLbvyNv2CaeHbl7mZLdGeaGqiVu0Je9sqqr % pepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs % 0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaai % aabeqaamaabaabauaakeaacuaH0oazgaqeaaaa!40DD!</EquationSource><EquationSource Format="TEX"><![CDATA[$$\bar \delta $$ -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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Literature Cited

  1. Agranovich, M. S., Elliptic singular integrodifferential operators, Usp. Mat. Nauk, 20 (5): 3–120 (1965).

    MathSciNet  MATH  Google Scholar 

  2. Agranovich, M. S., Positive boundary-value problems for certain first-order systems, Trudy Moskov. Mat. Obshch., 16: 3–24 (1967).

    MATH  Google Scholar 

  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. 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. Vainberg, B. R. and Grushin, V. V., Uniformly-nonelliptic problems (I), Mat. Sb., 72 (4): 602–636 (1967).

    Google Scholar 

  6. Vainberg, B. R. and Grushin, V. V., Uniformly-nonelliptic problems (II), Mat. Sb., 73 (1): 126–154 (1967).

    Google Scholar 

  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).

    MathSciNet  Google Scholar 

  8. Volevich, L. R., On general systems of differential equations, Dokl. Akad. Nauk SSSR, 132 (1): 20–23 (1960).

    Google Scholar 

  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. Volevich, L. R., Solvability of boundary-value problems for general elliptic sys- tems, Mat. Sb., 68 (3): 373–416 (1965).

    Google Scholar 

  11. Volevich, L. R., Local properties of inhomogeneous pseudodifferential operators, Trudy Moskov. Mat. Obsch., 16: 51–98 (1967).

    MATH  Google Scholar 

  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. Galakhov, M. A., Nonclassical boundary-value problems for symmetric linear first-order partial differential systems, Differentsial’nye Uravneniya, 1 (12): 1620–1627 (1965).

    MathSciNet  Google Scholar 

  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. 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. 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. 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. Gorfn, E. A., Solvability of the Cauchy problem, Vest. Moskov. Univ., Mat. Mekh., No. 4, pp. 6–12 (1965).

    Google Scholar 

  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).

    MathSciNet  Google Scholar 

  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).

    MathSciNet  MATH  Google Scholar 

  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. 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. Zolotareva, E. V., The Dirichlet problem for some second-order elliptic systems, Differentsial’nye Uravneniya, 3 (1): 59–68 (1967).

    MathSciNet  MATH  Google Scholar 

  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. 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).

    MathSciNet  MATH  Google Scholar 

  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).

    MATH  Google Scholar 

  27. Kuz’min, E. M., The Dirichlet problem for elliptic systems in a space, Differentsial’nye Uravneniya, 3 (1): 155–157 (1967).

    MathSciNet  MATH  Google Scholar 

  28. Lbdnev, N. A., A new method of solving partial differential equations, Mat. Sb., 22: 205–266 (1948).

    Google Scholar 

  29. Lopatinskii, Ya. B., Properties of linear differential operators, Mat. Sb., 17 (2): 267–285 (1945).

    Google Scholar 

  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. 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. 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. Nguen Tkhya Khop, The Dirichlet problem for strongly-connected elliptic systems, Dokl. Akad. Nauk SSSR, 171 (2): 292–295 (1966).

    Google Scholar 

  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).

    MathSciNet  MATH  Google Scholar 

  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).

    MathSciNet  Google Scholar 

  36. Palamodov, V. P., Systems of differential equations with constant coefficients, Dokl. Akad. Nauk SSSR, 148 (3): 523–526 (1963).

    MathSciNet  Google Scholar 

  37. Palamodov, V. P., Linear Differential Operators with Constant Coefficients, Nauka, Moscow (1967).

    Google Scholar 

  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).

    MathSciNet  MATH  Google Scholar 

  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. Palamodov, V. P., Differential operators in coherent analytic sheaves, Mat. Sb., 70 (3): 390–422 (1966).

    Google Scholar 

  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. Parasyuk, L. S., Generalized fundamental solution of elliptic systems of differential equations with discontinuous coefficients, Ukrain. Mat. Zh., 18 (4): 124–126 (1966).

    MATH  Google Scholar 

  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. Skripnik, I. V., The a-Neumann problem, Dopovidi Akad. Nauk Ukrain. RSR, No. 3, pp. 295–299 (1966).

    MathSciNet  Google Scholar 

  45. Solomyak, M. Z., On first-order linear elliptic systems, Dokl. Akad. Nauk SSSR, 150 (1): 48–51 (1963).

    Google Scholar 

  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).

    MathSciNet  MATH  Google Scholar 

  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).

    MathSciNet  Google Scholar 

  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. 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. 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. Tovmasyan, N. E., General boundary-value problem for second-order elliptic systems with constant coefficients (I), Differentsial’nye Uravneniya, 2 (1): 3–23 (1966).

    MathSciNet  Google Scholar 

  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).

    MathSciNet  Google Scholar 

  53. Fridlender, V. R., Polynomial matrices and systems of partial differential equations, Izv. Vuzov, Mat., No. 5, pp. 118–123 (1966).

    Google Scholar 

  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. Hörmander, L., An Introduction to Complex Analysis in Several Variables, Van Nostrand, Princeton, N. J. (1966).

    Google Scholar 

  56. Andreotti, A. and Grauert, H., Finiteness theorems for the cohomologies of complex spaces [in French], Bull. Soc. Math. France, 90: 193–259 (1962).

    MathSciNet  MATH  Google Scholar 

  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).

    MATH  Google Scholar 

  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).

    MathSciNet  MATH  Google Scholar 

  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).

    MathSciNet  MATH  Google Scholar 

  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. 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).

    MathSciNet  MATH  Google Scholar 

  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).

    MathSciNet  MATH  Google Scholar 

  63. Brenner, P., The Cauchy problem for symmetric hyperbolic systems in Lp, Math. Scandinay., 19 (1): 27–37 (1966).

    MATH  Google Scholar 

  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).

    MathSciNet  MATH  Google Scholar 

  65. Cenkl, B., The elliptic differential operators, Comment. Math. Univ. Carolinae, 8 (2): 175–197 (1967).

    MathSciNet  MATH  Google Scholar 

  66. Cenkl, B., Vanishing theorem for an elliptic differential operator, J. Differential Geom., 1: 381–418 (1967).

    MathSciNet  MATH  Google Scholar 

  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. 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. 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. 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).

    MathSciNet  MATH  Google Scholar 

  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).

    MathSciNet  Google Scholar 

  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).

    MathSciNet  Google Scholar 

  73. Douglis, A. and Nirenberg, L., Interior estimates for elliptic systems of partial equations, Commun. Pure Appl. Math., 8 (4): 503–538 (1955).

    MathSciNet  MATH  Google Scholar 

  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. 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. Ehrenpreis, L., Guillemin, V. W. and Sternberg, S., On Spencer’s estimate for 5-Poincaré, Ann. Math., 83 (1): 128–138 (1965).

    MathSciNet  Google Scholar 

  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).

    MathSciNet  Google Scholar 

  78. Friedrichs, K. O. and Lax, P. D., Boundary-value problems for first-order operators, Commun. Pure Appl. Math., 18: 355–388 (1965).

    MathSciNet  MATH  Google Scholar 

  79. Fuglede, B., A priori inequalities connected with systems of partial differential equations, Acta Math., 105 (1961).

    Google Scholar 

  80. Garabedian, P. R. and Spencer, D. C., Complex boundary problems, Trans. Amer. Math. Soc., 73: 223–242 (1952).

    MathSciNet  MATH  Google Scholar 

  81. Garding, L., Energy inequalities for hyperbolic systems, in: Differential Analysis, Oxford Univ. Press, London (1964), pp. 209–225.

    Google Scholar 

  82. Garding, L., A variation on the Cauchy majorizing method [in French], Acta Math., 114 (1–2): 143–158 (1965).

    MathSciNet  MATH  Google Scholar 

  83. Goldschmidt, H., Existence theorems for analytic linear partial differential equations, Ann. Math., 86 (2): 246–270 (1967).

    MathSciNet  MATH  Google Scholar 

  84. Guillemin, V. W., Some algebraic results concerning the characteristics of over-determined partial differential equations, Amer. J. Math., 90 (1): 270–284 (1968).

    MathSciNet  MATH  Google Scholar 

  85. Guillemin, V. W. and Sternberg, S., An algebraic model of transitive differential geometry, Bull. Amer. Math. Soc., 70 (1): 16–47 (1964).

    MathSciNet  MATH  Google Scholar 

  86. Hörmander, L., The Frobenius-Nirenberg theorem, Arkiv Mat., 5 (5): 425–432 (1964).

    Google Scholar 

  87. Hörmander, L., L2-estimates and existence theorems for the 5 operator, Acta Math., 113 (1–2): 89–152 (1965).

    MathSciNet  MATH  Google Scholar 

  88. Hörmander, L., Pseudodifferential operators and nonelliptic boundary problems, Ann. Math., 83 (1): 129–209 (1966).

    MATH  Google Scholar 

  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. Kodaira, K., On a differential geometric method in the theory of analytic stacks, Proc. Nat. Acad. Sci. USA, 39: 1268–1273 (1953).

    MathSciNet  MATH  Google Scholar 

  91. Kohn, J. J., Solution of the -(5.-Neumann problem on strongly-pseudoconvex manifolds, Proc. Nat. Acad. Sci. USA, 47: 1198–1202 (1961).

    MathSciNet  MATH  Google Scholar 

  92. Kohn, J. J., Harmonic integralsonstrongly-pseudoconvex manifolds (I), Ann. Math., 78 (1): 112–148 (1963).

    MathSciNet  MATH  Google Scholar 

  93. Kohn, J. J., Harmonic integrals on strongly-pseudoconvex manifolds (II), Ann. Math., 79: 450–472 (1964).

    MathSciNet  MATH  Google Scholar 

  94. Kohn, J. J., Boundaries of complex manifolds, Proc. Conf. Complex Analysis, Minneapolis, 1964, Springer-Verlag, Berlin (1965), pp. 81–94.

    Google Scholar 

  95. Kohn, J. J., Differential operators on manifolds with boundary, in: Differential Analysis, Oxford Univ. Press, London (1964), pp. 57–63.

    Google Scholar 

  96. Kohn, J. J., Differential complex, Proc. Internat. Congr. Mathematicians, 1966, Mir, Moscow (1968), pp. 402–409.

    Google Scholar 

  97. Kohn, J. J. and Nirenberg, L., Noncoercive boundary-value problems, Commun. Pure Appl. Math., 18: 443–492 (1965).

    MathSciNet  MATH  Google Scholar 

  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).

    MathSciNet  MATH  Google Scholar 

  99. Kohn, J. J., and Spencer, D. C., Complex Neumann problems, Ann. Math., 66: 98–140 (1957).

    MathSciNet  Google Scholar 

  100. Komatsu, H., Resolutions by hyperfunctions of sheaves of solutions of differential equations with constant coefficients, Math. Ann., 176 (1): 77–86 (1968).

    MathSciNet  MATH  Google Scholar 

  101. Kopacek, J., The Cauchy problem for linear hyperbolic systems in Lp, Comment. Math. Univ. Carolinae, 8 (3): 459–462 (1968).

    MathSciNet  Google Scholar 

  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. Kuranishi, M., Lectures on Involutive Systems of Partial Differential Equations (Publ. Soc. Mat. STo Paulo), STo Paulo (1967), 77 pages.

    Google Scholar 

  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. Lax, P. D. and Phillips, R. S., Local boundary conditions for dissipative symmetric linear differential operators, Commun. Pure Appl. Math., 13: 427–455 (1960).

    MathSciNet  MATH  Google Scholar 

  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. 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. 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. 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. 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. 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. Malgrange, B., Some convexity problems for differential operators with constant coefficients, Sem. Leray, Collège, de France, Paris (1962/63), pp. 190223.

    Google Scholar 

  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. 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. 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. Matsuura, S., On general systems of partial differential operators with constant coefficients, J. Math. Soc. Japan, 13 (1): 94–103 (1961).

    MathSciNet  MATH  Google Scholar 

  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).

    MathSciNet  MATH  Google Scholar 

  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).

    MathSciNet  MATH  Google Scholar 

  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).

    MathSciNet  MATH  Google Scholar 

  120. Morrey, C. B., The analytic embedding of abstract real-analytic manifolds, Ann. Math., 68: 159–201 (1958).

    MathSciNet  MATH  Google Scholar 

  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. 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).

    MathSciNet  MATH  Google Scholar 

  123. Newlander, A. and Nirenberg, L., Complex-analytic coordinates in almost-complex manifolds, Ann., Math., 65: 391–404 (1957).

    MathSciNet  MATH  Google Scholar 

  124. Nirenberg, L., A complex Frobenius theorem, Sem. Analytic Functions, Vol. 1, Inst. Adv. Study, Princeton (1957), pp. 172–189.

    Google Scholar 

  125. Persson, J., A boundary problem for analytic linear systems with data on intersecting hyperplanes, Math. Scandinay., 14 (1): 106–110 (1964).

    MathSciNet  MATH  Google Scholar 

  126. Phillips, R. S. and Sarason, L., Singular symmetric positive first-order differential operators, J. Math., Mech., 15 (2): 235–271 (1956).

    MathSciNet  Google Scholar 

  127. Schaefer, P. W., On the Cauchy problem for an elliptic system, Arch. Rat. Mech. Anal., 20 (5); 391–412 (1965).

    MathSciNet  MATH  Google Scholar 

  128. Schechter, M., Systems of partial differential equations in a half-space, Commun. Pure Appt. Math., 17 (4): 423–434 (1964).

    MathSciNet  MATH  Google Scholar 

  129. Schechter, M., Deformation of structures on manifolds defined by transitive continuous pseudogroups (I), Ann. Math., 76; 306–398 (1962).

    Google Scholar 

  130. Schechter, M., Deformation of structures on manifolds defined by transitive continuous pseudogroups (II), Ann. Math., 76: 399–445 (1962).

    Google Scholar 

  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. 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. 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. Spencer, D. C., D.formation of structures on manifolds defined by transitive continuous pseudogroups (II), Ann. Math., 81: 389–450 (1962).

    Google Scholar 

  135. Sweeney, W. J., The 6-Poincaré estimate, Pacific J. Math., 20 (3): 559–570 (1967).

    MathSciNet  MATH  Google Scholar 

  136. Sweeney, W. J., A noncompact Dirichlet norm, Proc. Nat. Acad. Sci. USA, 58 (6): 2193–2195 (1967).

    MATH  Google Scholar 

  137. Sweeney, W. J., The D-Neumann problem, Acta Math., 120 (3–4): 223–277 (1968).

    MathSciNet  MATH  Google Scholar 

  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).

    MathSciNet  MATH  Google Scholar 

  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. 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).

    MATH  Google Scholar 

  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).

    MATH  Google Scholar 

  142. Zerner, M., Singular solutions of partial differential equations [in French], Bull. Soc. Math. France, 91: 302–226 (1963).

    MathSciNet  Google Scholar 

Download references

Authors
  1. V. P. Palamodov
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. V.A. Steklov Mathematics Institute, Academy of Sciences of the USSR, Moscow, USSR

    R. V. Gamkrelidze

Rights and permissions

Reprints and permissions

Copyright information

© 1971 Springer Science+Business Media New York

About this chapter

Cite this chapter

Palamodov, V.P. (1971). Systems of Linear Differential Equations. In: Gamkrelidze, R.V. (eds) Mathematical Analysis. Progress in Mathematics, vol 10. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-1589-7_1

Download citation

  • DOI: https://doi.org/10.1007/978-1-4757-1589-7_1

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4757-1591-0

  • Online ISBN: 978-1-4757-1589-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation