Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)

 
 

 

Some recent developments in the theory of partial differential equations


Author: Charles B. Morrey Jr.
Journal: Bull. Amer. Math. Soc. 68 (1962), 279-297
DOI: https://doi.org/10.1090/S0002-9904-1962-10765-4
MathSciNet review: 1566193
Full-text PDF

References | Additional Information

References [Enhancements On Off] (What's this?)

  • 1. S. Agmon, Maximum theorems for solutions of higher order elliptic equations, Bull. Amer. Math. Soc. 66 (1960), 77-80. MR 124618
  • 2. S. Agmon, A Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math. 12 (1959), 623-727. MR 125307
  • 3. N. Aronszajn, Sur l'unicité du prolongement des solutions des équations aux dérivées partielles elliptiques du second ordre, C. R. Acad. Sci. Paris 242 (1956), 723-725. MR 76155
    3a. N. Aronszajn, A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order, J. Math. Pures. Appl. 36 (1957), 235-249. MR 92067

  • 4. N. Aronszajn, A. Krzywicki and J. Szarski, Unique continuation theorem for exterior differential forms on Riemannian manifolds, ONR Tech. Rep. No. 25, Univ. of Kansas, Lawrence, Kans., 1960; to appear in Arkiv. for Math. MR 140031
  • 5. N. Aronzajn, Associated spaces, interpolation theorems and the regularity of the solutions of differential problems, Proc. Sympos. Pure Math. Vol. 4, pp. 23-32, Amer. Math. Soc., Providence, R. I., 1962. MR 140854
    5a. L. Bers, Non-linear elliptic equations without non-linear entire solutions, J. Rational Mech. Anal. 3 (1954), 767-787. MR 67313

  • 6. F. E. Browder, The Dirichlet problem for linear elliptic equations of arbitrary even order with variable coefficients, Proc. Nat. Acad. Sci. U.S.A. 38 (1952), 230-235. MR 49442
  • 7. F. E. Browder, A priori estimates for elliptic and parabolic equations, Proc. Sympos. Pure Math. Vol. 4, pp. 73-81, Amer. Math. Soc. Providence, R. I., 1962. MR 131066
    7a F. E. Browder, On the spectral theory of elliptic differential operators. I, Math. Ann. 142 (1961), 22-130. MR 209909

  • 8. S. Bochner, Analytic mapping of compact Riemann spaces into Euclidean space, Duke Math. J. 3 (1937), no. 2, 339–354. MR 1545992, https://doi.org/10.1215/S0012-7094-37-00326-0
  • 9. A. P. Calderón, Uniqueness in the Cauchy problem for partial differential equations, Amer. J. Math. 80 (1958), 16-36. MR 104925
  • 10. A. P. Calderón, Lebesgue spaces of differentiable functions, Proc. Sympos. Pure Math. Vol. 4, pp. 33-49, Amer. Math. Soc., Providence, R. I., 1962. MR 143037
  • 11. A. P. Calderón and A. Zygmund, Singular integral operators and differential equations, Amer. J. Math. 79 (1957), 901-921. MR 100768
  • 12. T. Carleman, Sur les systèmes linéaires aux dérivées partielles du premier ordre à deux variables, C.R. Acad. Sci. Paris 197 (1933), 471-474.
  • 13. T. Carleman, Sur un problème d'unicité pour les systèmes d'équations aux dérivées partielles à deux variables indépendentes, Archiv. f. Mat. Fys. o Astr. (26B) 17 (1939), 1-9.
  • 14. P. J. Cohen, The non-uniqueness of the Cauchy problem, Tech. Rept. No. 93 (Dec. 30, 1960), Appl. Math, and Stat. Lab., Stanford University.
    14a. P. J. Cohen, (to appear).

  • 15. H. O. Cordes, Ueber die Bestimmtheit der Lösungen elliptischer Differentialgleichungen durch Anfangsvorgaben, Nachr, Akad. Wiss. Göttingen, Math.-Phys. Kl. (IIa) No. 11 (1956), 239-253. MR 86237
  • 16. H. O. Cordes, Zero order a priori estimates for solutions of elliptic differential equations, Proc. Sympos. Pure Math. Vol. 4, pp. 157-166, Amer. Math. Soc., Providence, R. I., 1962. MR 146511
  • 17. E. Di Giorgi, Un esempio di non-unicità délia soluzione del problema di Cauchy, relativo ad una equazione differenziale lineare a derivate parziali di tipo parabolico, Rend. Mat. e Appl. (5) 14 (1955), 382-387. MR 70028
  • 18. E. Di Giorgi, Sulla differenziabilità e l'analyticità delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino Cl. Sci. Fis. Mat. Nat. (3) 3 (1957), 25-43. MR 93649
  • 19. A. Douglis and L. Nirenberg, Interior estimates for elliptic systems of partial differential equations, Comm. Pure Appl. Math. 8 (1955), 503-538. MR 75417
  • 20. G. F. D. Duff, Partial differential equations, Math. Expositions No. 9, Toronto Univ. Press, Toronto, 1956. MR 78550
    20a. R. Finn, On equations of minimal surface type, Ann. of Math. (2) 60 (1954), 367-416. MR 66533

    20b. R. Finn, On a problem of type with application to elliptic partial differential equations, J. Rational Mech. Anal. 3 (1954), 789-799. MR 67314

  • 21. A. Friedman, On the regularity of the solutions of non-linear elliptic and parabolic systems of partial differential equations, J. Math. Mech. 7 (1958), 43-59. MR 118970
  • 22. K. O. Friedrichs, Symmetric positive linear differential equations, Comm. Pure Appl. Math. 11 (1958), 333-418. MR 100718
  • 23. L. Gårding, Le problème de Dirichlet pour les équations aux dérivées partielles elliptiques linéaires dans les domaines bornés, C. R. Acad. Sci. Paris 233 (1951), 1554-1556. MR 49441
  • 24. L. Gårding, Dirichlet's problem for linear elliptic partial differential equations, Math. Scand. 1 (1953), 55-72. MR 64979
  • 25. L. Gårding, Cauchy's problem for hyperbolic equations, Dept. of Math., Univ. of Chicago, Chicago, Ill., 1958.
  • 26. I. V. Gel'man, The minimum problem for a non-linear functional, Leningrad. Gos. Ped. Inst. Uč. Zap. 166 (1958), 255-263. MR 102031
  • 27. H. Grauert, On Levi's problem and the imbedding of real-analytic manifolds, Ann. of Math. (2) 68 (1958), 460-472. MR 98847
  • 28. L. Hörmander, On the theory of general partial differential operators, Acta Math. 94 (1955), 160-248. MR 76151
  • 29. L. Hörmander, Differential equations without solutions, Math. Ann. 140 (1960), 169-173. MR 147765
    29a. L. Hörmander, On the uniqueness of the Cauchy problem. II, Math. Scand. 7 (1959), 177-190. MR 121569

    29b. H. Jenkins, On quasi-linear elliptic equations which arise from variational problems, J. Math. Mech. 10 (1961), 705-727. MR 126741

  • 30. J. J. Kohn, Solution of the {$\bar \partial $}-Neumann problem on strongly pseudo-convex manifolds, Proc. Nat. Acad. Sci. U.S.A. 47 (1961), 1198-1202. MR 133596
  • 31. O. A. Ladyženskaya and N. N. Uralt'seva, On the smoothness of weak solutions of quasi-linear equations in several variables and of variational problems, Comm. Pure Appl. Math. 14 (1961), 481-495. MR 149076
    31a. O. A. Ladyženskaya and N. N. Uralt'seva, Quasi-linear elliptic equations and variational problems with many independent variables, Russian Math. Surveys, London Math. Soc. (16) 1 (1961), 17-91.

  • 32. E. M. Landis, An example of non-uniqueness of solutions of Cauchy's problem for a system of the form ∂u, Mat. Sb. (N.S.) 27 (69) (1950), 319-323. MR 37985
  • 33. P. Lax, On Cauchy's problem for hyperbolic equations and the differentiability of the solutions of elliptic equations, Comm. Pure Appl. Math. 8 (1955), 615-633. MR 78558
    33a. P. D. Lax and A. N. Milgram, Parabolic equations, Ann. of Math. Studies No. 33, Princeton Univ. Press, Princeton, N. J., 1954; pp. 167-190, especially p. 169. MR 67317

  • 34. J. Leray, Majoration des dérivées secondes des solutions d'un problème de Dirichlet, J. Math. Pures Appl. (9) 17 (1938), 89-104.
  • 35. J. Leray, Discussion d'un problème de Dirichlet, J. Math. Pures Appl. (9) 18 (1939), 249-284. MR 1094
  • 36. J. Leray, Lectures on hyperbolic equations with variable coefficients, Institute for Advanced Studies, Princeton, N. J., 1952.
  • 37. B. Levi, Sul principio di Dirichlet, Rend. Cire. Nat. Palermo 22 (1906), 293-359.
  • 38. H. Lewy, An example of a smooth linear partial differential equation without solution, Ann. of Math. (2) 66 (1957), 155-158. MR 88629
  • 39. J. L. Lions, Sur certains problèmes différentiels non linéaires, C. R. Acad. Sci. Paris 252 (1961), 657-659. MR 133606
  • 40. J. L. Lions, Equations différentielles opérationnelles et problèmes aux limites, Die Grundlehren der mathematischen Wissenschaften, Springer, Berlin, 1961. MR 153974
  • 41. B. Malgrange, Plongement des variétés analytiques réelles, Bull. Soc. Math. France 85 (1957), 101-113. MR 94829
  • 42. C. Miranda, Equazioni alle derivate parziali di tipo ellittico, Ergebnisse der Mathematik und ihrer Grenzgebiete (N.F.), Heft 2, Springer, Berlin, 1955. MR 87853
  • 43. C. Miranda, Teorema del massimo modulo e teorema di esistenza e di unicità per ü problema di Dirichlet relativo aile equazioni ellittiche in due variabili, Ann. Mat. Pura Appl. 46 (1958), 265-311. MR 124615
  • 44. Charles B. Morrey Jr., On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc. 43 (1938), no. 1, 126–166. MR 1501936, https://doi.org/10.1090/S0002-9947-1938-1501936-8
  • 45. C. B. Morrey, Jr., Existence and differentiability theorems for the solutions of variational problems, Bull. Amer. Math. Soc. 46 (1940), 439-458. MR 2473
  • 46. C. B. Morrey, Jr., Multiple integral problems in the calculus of variations and related topics, Univ. California Publ. Math. (N.S.) 1 (1943), 1-130. MR 11537
  • 47. C. B. Morrey, Jr., The problem of Plateau on a Riemannian manifold, Ann. of Math, (2) 49 (1948), 807-851. MR 27137
  • 48. C. B. Morrey, Jr., Quasi-convexity and the lower semicontinuity of multiple integrals, Pacific J. Math. 2 (1952), 25-53. MR 54865
  • 49. C. B. Morrey, Jr., Second order elliptic systems of differential equations, Ann. of Math. Studies No. 33, Princeton Univ. Press, Princeton, N. J., 1954; pp. 101-159. MR 68091
  • 50. C. B. Morrey, Jr., A variational method in the theory of harmonic integrals. II, Amer. J. Math. 78 (1956), 137-170. MR 87765
  • 51. C. B. Morrey, Jr., On the analyticity of the solutions of analytic non-linear elliptic systems of partial differential equations. I, II, Amer. J. Math. 80 (1958), 198-237. MR 106336
  • 52. C. B. Morrey, Jr., Elliptic differential equations with Hölder continuous coefficients, Amer. Math. Soc. Notices 5 (1958), 466.
  • 53. C. B. Morrey, Jr., The analytic embedding of abstract real-analytic manifolds, Ann. of Math. (2) 68 (1958), 159-201. MR 99060
  • 54. C. B. Morrey, Jr., Second order elliptic equations in several variables and Hölder continuity, Math. Z. 72 (1959), 146-164. MR 120446
  • 55. C. B. Morrey, Jr., Multiple integral problems in the calculus of variations and related topics, Ann. Scuola Norm. Pisa (III) 14 (1960), 1-61. MR 115117
  • 55. C. B. Morrey, Jr., Existence and differentiability theorems for variational problems for multiple integrals, Partial differential equations and continuum mechanics, Univ. of Wisconsin Press, Madison, Wis., 1961. MR 121690
  • 57. C. B. Morrey, Jr. and J. Eells, Jr., A variational method in the theory of harmonic integrals. I, Ann. of Math. (2) 63 (1956), 91-128. MR 87764
  • 58. C. B. Morrey, Jr. and L. Nirenberg, On the analyticity of the solutions of linear elliptic systems of partial differential equations, Comm. Pure Appl. Math. 10 (1957), 271-290. MR 89334
  • 59. L. Moser, A new proof of di Giorgi's theorem concerning the regularity problem for elliptic differential equations, Comm. Pure Appl. Math. 13 (1960), 457-468. MR 170091
  • 60. A. Myskis, On the method of A. Haar in a question of the uniqueness of the solution of the problem of Cauchy for a system of partial differential equations, Dokl. Akad. Nauk. SSSR (N.S.) 58 (1947), 21-24. MR 23427
  • 61. J. Nash, Continuity of the solutions of parabolic and elliptic equations, Amer. J. Math. 80 (1958), 931-954. MR 100158
  • 62. A. Newlander and L. Nirenberg, Complex analytic coordinates in almost complex manifolds, Ann. of Math. (2) 65 (1957), 391-404. MR 88770
  • 63. J. Nieto, Eine Charakterisierung der elliptischen Differentialoperatoren, Math. Ann. 141 (1960), 22-42. MR 116138
  • 64. L. Nirenberg, On non-linear elliptic partial differential equations and Hölder continuity, Comm. Pure Appl. Math. 6 (1953), 103-156. MR 64986
  • 65. L. Nirenberg, Remarks on strongly elliptic partial differential equations, Comm. Pure Appl. Math. 8 (1955), 648-674. MR 75415
  • 66. L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Pisa (III) 13 (1959), 1-48. MR 109940
  • 67. L. E. Payne, Inequalities for eigenvalues of membranes and plates, J. Rational Mech. Anal. 4 (1955), 517-529. MR 70834
  • 68. L. E. Payne and H. F. Weinberger, New bounds in harmonic and biharmonic problems, J. Math. Phys. 33 (1955), 291-307. MR 68683
  • 69. L. E. Payne and H. F. Weinberger, Lower bounds for frequencies of elastically supported membranes and plates, J. Soc. Indust. Appl. Math. 5 (1957), 171-182. MR 92431
  • 70. L. E. Payne and A. Weinstein, Capacity, virtual mass, and generalized symmetrization, Pacific J. Math. 2 (1952), 633-641. MR 50738
  • 71. I. Petrovsky, Sur l'analyticité des solutions des systèmes d'équations différentielles, Rec. Mat. N.S. Mat. Sbornik 5(47) (1939), 3-70.
  • 72. I. Petrovsky, Lectures on partial differential equations. I (Translation from the 1950 Russian edition), Interscience, New York, 1954. MR 65760
  • 73. A. Plis, The problem of uniqueness for the solution of a system of partial differential equations, Bull. Acad. Polon. Sci. Cl. III 2 (1954), 55-57. MR 64279
    73a. A. Plis, A smooth linear elliptic differential equation without any solution in a sphere, Comm. Pure Appl. Math. 14 (1961), 599-617. MR 136846

  • 74. G. Polya and G. Szegö, Inequalities for the capacity of a condenser, Amer. J. Math. 67 (1945), 1-32. MR 11871
  • 75. P. C. Rosenbloom, Linear partial differential equations, Surveys in Applied Math., Vol. 5, New York, 1958.
  • 76. M. Schechter, General boundary value problems for elliptic partial differential equations, Comm. Pure Appl. Math. 12 (1959), 457-486. MR 125323
  • 77. M. Schechter, Various types of boundary conditions for elliptic equations, Comm. Pure Appl. Math. 13 (1960), 407-425. MR 125332
    77a. H. Seifert and W. Threlfall, Variationsrechnung im Grossen (Theorie von Marston Morse), Chelsea, New York, 1951.

  • 78. J. Serrin, On a fundamental theorem of the calculus of variations, A new definition of the integral for non-parametric problems in the calculus of variations, Acta Math. 102 (1959), 1-32. MR 108745
  • 79. J. Serrin, Dirichlet's principle in the calculus of variations, Proc. Sympos. Pure Math. Vol. 4, pp. 17-22, Amer. Math. Soc., Providence, R. I., 1962. MR 137012
    79a. J. Serrin, On the definition and properties of certain variational integrals, Trans. Amer. Math. Soc. 101 (1961), 139-167. MR 138018

  • 80. A. G. Sigalov, Two dimensional problems in the calculus of variations, Uspehi Mat. Nauk (N.S.) 6 (1951), 16-101; Amer. Math. Soc. Transl. No. 83 (1953). MR 43400
  • 81. A. G. Sigalov, Two dimensional problems of the calculus of variations in non-parametric form, Trudy Moskov. Mat. Obšč. 2 (1953), 201-233 (Russian); MR 15, 442. MR 58884
  • 82. G. I. Silova, Existence of an absolute minimum of multiple integrals of the calculus of variations, Dokl. Akad. Nauk SSSR (N.S.) 102 (1955), 699-702; MR 17, 46. MR 70861
  • 83. S. Sobolev, On a theorem of functional analysis, Mat. Sb. (N.S.) 4 (1938), 471-497.
  • 84. G. Stampacchia, Contributi alla regolarizzazione delle soluzioni dei problemi al contorno per equazioni del secondo ordine ellitiche, Ann. Scuola Norm. Sup. Pisa (III) 12 (1958), 223-244. MR 125313
  • 85. G. Stampacchia, Problemi al contorno ellittici, con dati discontinui, dotati di soluzioni hölderiane, Ann. Mat. Pura Appl. (IV) 51 (1960), 1-38. MR 126601
  • 86. L. Tonelli, Fondamenti del calcolo delle variazioni, Bologna, Zanichelli, 3 vols. 87. L. Tonelli, sur la semi-continuité des intégrales doubles du calcul des variations, Acta Math. 53 (1929), 325-346.
  • 88. L. Tonelli, L'estremo assoluto degli integrali doppi, Ann. Scuola Norm. Pisa (II) 3 (1933), 89-130.
  • 89. L. Van Hove, Sur l'extension de la condition de Legendre du calcul des variations aux intégrals multiples a plusieurs fonctions inconnues, Nederl. Akad. Wetensch. 50 (1947), 18-23. MR 20223
  • 90. M. I. Visik, On strongly elliptic systems of differential equations, Mat. Sb. (N.S.) 29 (71) (1951), 615-676. MR 49440
  • 91. A. Weinstein, Etude des spectres des équations aux dérivées partielles de la théorie des plaques élastiques, Mem. Sci. Math., Fasc. 88, Gauthier-Villars, Paris, 1937.
  • 92. L. C. Young, On generalized surfaces of finite topological type, Mem. Amer. Math. Soc. No. 17 (1955). MR 88671
  • 93. K. Yosida, Lectures on semi-group theory and its application to Cauchy's problem in partial differential equations, Tata Institute Lectures on Mathematics and Physics Math. 8.
    F. John, Plane waves and spherical means applied to partial differential equations, Interscience, New York, 1955. MR 75429

    S. Bergman, Kernel functions and elliptic differential equations in mathematical physics, Academic Press, New York, 1953. MR 54140

    C. Carathéodory, Variationsrechnung und partielle Differentialgleichungen erster Ordnung, Band I, 2te Aufl., B. G. Teubner, Leipzig, 1956. MR 89338

    J. L. Lions, Lectures on elliptic partial differential equations, Tata Institute of Fundamental Research, Lectures on Mathematics, No. 10. MR 265732

    J. L. Lions, Boundary value problems, Dept. of Math., Univ. of Kansas, Lawrence, Kans.

    L. Nirenberg, Existence theorems in partial differential equations. New York University, Inst. Math. Sci.

    L. Schwartz, Lectures on mixed problems in partial differential equations and representations of semigroups, Tata Institute of Fundamental Research, 1957.

    Contributions to the theory of partial differential equations, Ann. of Math. Studies No. 33, Princeton Univ. Press, Princeton, N. J., 1954.

    Proc. of the 1955 Berkeley Symposium(a)Lectures, Dept. of Math., Univ. of Kansas, Lawrence, Kans.

    Proc. of the 1955 Berkeley Symposium (b) Transactions, Comm. Pure Appl. Math. 9 (1956).

    Proc. Conference on Partial Differential Equations, Univ. of Kansas, Lawrence, Kans., 1954.

    Partial differential equations, Proc. Sympos. Pure Math. Vol. 4, American Mathematical Society, Providence, R. I., 1962 (1960 Berkeley Symposium).

    Partial differential equations and continuum mechanics, Math. Res. Center, Univ. of Wisconsin Press, Madison, Wis., 1961.

    L. Gårding, Some trends and problems in linear differential equations, Proc. Int. Congr. Math., 1958.

    G. Stampacchia, I problemi al contorno per le equazioni differenziali di tipo ellittico, Atti VI Congr. Un. Mat. Ital. (Naples, 1959), pp. 21-44, Edizione Cremonese, Rome 1960. MR 123807

    A. Weinstein, Bounds for eigenvalues and the method of intermediate problems, Partial Differential Equations and Continuum Mechanics, pp. 39-53, Math. Res. Center, Univ. of Wisconsin Press, Madison, Wis., 1961. MR 126068


Additional Information

DOI: https://doi.org/10.1090/S0002-9904-1962-10765-4

American Mathematical Society