V.A. Steklov’s work on equations of mathematical physics and development of his results in this field

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Abstract

This paper is an extended account of the author’s talk at the International Conference “Contemporary Problems of Mathematics, Mechanics, and Mathematical Physics” dedicated to the 150th anniversary of Vladimir Andreevich Steklov. Steklov’s main studies on the solvability of boundary value problems for equations of mathematical physics are briefly described, and the further development of this field of research is surveyed. The main attention is focused on the statements of the Dirichlet problem and the conditions on the domain and given functions under which solvability theorems are valid.

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References

  1. 1.
    Yu. A. Alkhutov, “L p-estimates of the solution of the Dirichlet problem for second-order elliptic equations,” Mat. Sb. 189(1), 3–20 (1998) [Sb. Math. 189, 1–17 (1998)].MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Yu. A. Alkhutov and V. A. Kondrat’ev, “Solvability of the Dirichlet problem for second-order elliptic equations in a convex region,” Diff. Uravn. 28(5), 806–818 (1992) [Diff. Eqns. 28, 650–662 (1992)].MathSciNetGoogle Scholar
  3. 3.
    O. V. Besov, V. P. Il’in, and S. M. Nikol’skii, Integral Representations of Functions and Imbedding Theorems (Nauka, Moscow, 1996) [in Russian].MATHGoogle Scholar
  4. 4.
    L. Carleson, “An interpolation problem for bounded analytic functions,” Am. J. Math. 80, 921–930 (1958).MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    L. Carleson, “Interpolations by bounded analytic functions and the corona problem,” Ann. Math., Ser. 2, 76, 547–559 (1962).MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    E. De Giorgi, “Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli regolari,” Mem. Accad. Sci. Torino, Cl. Sci., Fis. Mat. Nat., Ser. 3, 3, 25–43 (1957).MATHGoogle Scholar
  7. 7.
    V. Zh. Dumanyan, “Solvability of the Dirichlet problem for a general second-order elliptic equation,” Mat. Sb. 202(7), 75–94 (2011) [Sb. Math. 202, 1001–1020 (2011)].MathSciNetCrossRefGoogle Scholar
  8. 8.
    J. Fourier, Théorie analytique de la chaleur (Firmin Didot Père et Fils, Paris, 1822); Engl. transl.: The Analytical Theory of Heat (Dover Publ., New York, 1955).Google Scholar
  9. 9.
    D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order (Springer, Berlin, 1983).CrossRefMATHGoogle Scholar
  10. 10.
    G. Giraud, “Sur le problème de Dirichlet généralisé (deuxième mémoire),” Ann. Sci. Éc. Norm. Super., Sér. 3, 46, 131–245 (1929).MathSciNetMATHGoogle Scholar
  11. 11.
    G. Giraud, “Sur certains problèmes non linéaires de Neumann et sur certains non linéaires mixtes,” Ann. Sci. Éc. Norm. Super., Sér. 3, 49, 1–104 (1932).MathSciNetGoogle Scholar
  12. 12.
    G. Giraud, “Sur certains problèmes non linéaires de Neumann et sur certains non linéaires mixtes (suite),” Ann. Sci. Éc. Norm. Super., Sér. 3, 49, 245–309 (1932).MathSciNetMATHGoogle Scholar
  13. 13.
    A. K. Gushchin, “On the Dirichlet problem for a second-order elliptic equation,” Mat. Sb. 137(1), 19–64 (1988) [Math. USSR, Sb. 65, 19–66 (1990)].MATHGoogle Scholar
  14. 14.
    A. K. Gushchin, “On the interior smoothness of solutions to second-order elliptic equations,” Sib. Mat. Zh. 46(5), 1036–1052 (2005) [Sib. Math. J. 46, 826–840 (2005)].MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    A. K. Gushchin, “The Dirichlet problem for a second-order elliptic equation with an L p boundary function,” Mat. Sb. 203(1), 3–30 (2012) [Sb. Math. 203, 1–27 (2012)].MathSciNetCrossRefGoogle Scholar
  16. 16.
    A. K. Gushchin, “L p-estimates for solutions of second-order elliptic equation Dirichlet problem,” Teor. Mat. Fiz. 174(2), 243–255 (2013) [Theor. Math. Phys. 174, 209–219 (2013)].MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    A. K. Gushchin and V. P. Mikhailov, “On the existence of boundary values of solutions of an elliptic equation,” Mat. Sb. 182(6), 787–810 (1991) [Math. USSR, Sb. 73, 171–194 (1992)].MATHGoogle Scholar
  18. 18.
    N. M. Günter, Potential Theory, and Its Applications to Basic Problems of Mathematical Physics (Gostekhizdat, Moscow, 1953; Frederick Ungar, New York, 1967).Google Scholar
  19. 19.
    D. Hilbert, “Über das Dirichletsche Prinzip,” Jahresber. Dtsch. Math.-Ver. 8, 184–188 (1900).MATHGoogle Scholar
  20. 20.
    E. Hopf, “Elementare Bemerkungen über die Lösungen partieller Differentialgleichungen zweiter Ordnung vom elliptichen Typus,” Sitzungsber. Preuss. Akad. Wiss., Phys.-Math. Kl., 147–152 (1927).Google Scholar
  21. 21.
    E. Hopf, “Über den funktionalen, insbesondere den analytischen Charakter der Lösungen elliptischer Differentialgleichungen zweiter Ordnung,” Math. Z. 34, 194–233 (1932).MathSciNetCrossRefGoogle Scholar
  22. 22.
    O. Hölder, “Beiträge zur Potentialtheorie,” Inaugural-Diss. (Univ. Tübingen, 1882).Google Scholar
  23. 23.
    L. Hörmander, “L p estimates for (pluri-)subharmonic functions,” Math. Scand. 20, 65–78 (1967).MathSciNetMATHGoogle Scholar
  24. 24.
    M. V. Keldysh, “On the solvability and stability of the Dirichlet problem,” Usp. Mat. Nauk, No. 8, 171–231 (1941) [Am. Math. Soc. Transl., Ser. 2, 51, 1–73 (1966)].Google Scholar
  25. 25.
    V. I. Kondrashov, “Certain properties of functions in the space L pv,” Dokl. Akad. Nauk SSSR 48(8), 563–566 (1945).Google Scholar
  26. 26.
    V. A. Kondrat’ev and E. M. Landis, “Qualitative theory of second order linear partial differential equations,” in Partial Differential Equations-3 (VINITI, Moscow, 1988), Itogi Nauki Tekh., Ser.: Sovrem. Probl. Mat., Fundam. Napravl. 32, pp. 99–215; Engl. transl. in Partial Differential Equations III (Springer, Berlin, 1991), Encycl. Math. Sci. 32, pp. 87–192.Google Scholar
  27. 27.
    A. Korn, Lehrbuch der Potentialtheorie. Allgemeine Theorie des Potentials und der Potentialfunctionen im Raume (Ferd. Dümmler, Berlin, 1899).MATHGoogle Scholar
  28. 28.
    A. Korn, Über Minimalflächen, deren Randkurven wenig von ebenen Kurven abwiechen (Königl. Akad. Wiss., Berlin, 1909), Abh. K. Preuss. Akad. Wiss., Phys.-Math. Kl., Anhang 2.Google Scholar
  29. 29.
    A. Korn, “Zwei Anwendungen der Methode der sukzessiven Annäherungen,” in Schwarz-Festschrift (Springer, Berlin, 1914), pp.215–229.Google Scholar
  30. 30.
    O. A. Ladyzhenskaya and N. N. Ural’tseva, Linear and Quasilinear Elliptic Equations (Nauka, Moscow, 1964; Academic, New York, 1968); 2nd ed. (Nauka, Moscow, 1973).MATHGoogle Scholar
  31. 31.
    H. Lebesgue, “Sur le problème de Dirichlet,” Rend. Circ. Mat. Palermo 24, 371–402 (1907).CrossRefMATHGoogle Scholar
  32. 32.
    A. Liapounoff, “Sur certaines questions qui se rattachent au problème de Dirichlet,” J. Math. Pures Appl., Sér. 5, 4, 241–311 (1898).MATHGoogle Scholar
  33. 33.
    J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications (Springer, Berlin, 1972–1973), Vols. 1–3.CrossRefGoogle Scholar
  34. 34.
    J. E. Littlewood and R. E. A. C. Paley, “Theorems on Fourier series and power series,” J. Lond. Math. Soc. 6, 230–233 (1931).MathSciNetCrossRefGoogle Scholar
  35. 35.
    J. E. Littlewood and R. E. A. C. Paley, “Theorems on Fourier series and power series. II, III,” Proc. Lond. Math. Soc., Ser. 2, 42, 52–89 (1936); 43, 105–126 (1937).Google Scholar
  36. 36.
    L. A. Lyusternik, “Über einige Anwendungen der direkten Methoden in Variationsrechnung,” Mat. Sb. 33(2), 173–201 (1926).MATHGoogle Scholar
  37. 37.
    V. G. Maz’ya, Sobolev Spaces (Leningr. Gos. Univ., Leningrad, 1985; Springer, Berlin, 1985).Google Scholar
  38. 38.
    V. P. Mikhailov, “Dirichlet’s problem for a second-order elliptic equation,” Diff. Uravn. 12(10), 1877–1891 (1976) [Diff. Eqns. 12, 1320–1329 (1977)].Google Scholar
  39. 39.
    V. P. Mikhailov, Partial Differential Equations (Nauka, Moscow, 1983).Google Scholar
  40. 40.
    V. P. Mikhailov and A. K. Gushchin, Additional Chapters of the Course “Equations of Mathematical Physics” (Steklov Math. Inst., Moscow, 2007), Lekts. Kursy NOTs 7.Google Scholar
  41. 41.
    J. Moser, “A new proof of de Giorgi’s theorem concerning the regularity problem for elliptic differential equations,” Commun. Pure Appl. Math. 13(3), 457–468 (1960).CrossRefMATHGoogle Scholar
  42. 42.
    J. F. Nash, “Continuity of solutions of parabolic and elliptic equations,” Am. J. Math. 80, 931–954 (1958).MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    J. Nečas, “On the solutions of second order elliptic partial differential equations with unbounded Dirichlet integral,” Czech. Math. J. 10, 283–298 (1960).Google Scholar
  44. 44.
    O. A. Oleinik, “On the Dirichlet problem for elliptic equations,” Mat. Sb. 24(1), 3–14 (1949).MathSciNetGoogle Scholar
  45. 45.
    O. Perron, “Eine neue Behandlung der ersten Randwertaufgabe für Δu = 0,” Math. Z. 18, 42–54 (1923).MathSciNetCrossRefMATHGoogle Scholar
  46. 46.
    I. G. Petrovsky, “New proof of the existence of a solution to the Dirichlet problem by a finite difference method,” Usp. Mat. Nauk, No. 8, 161–170 (1941).MATHGoogle Scholar
  47. 47.
    I. G. Petrovsky, Lectures on Partial Differential Equations (Gostekhizdat, Moscow, 1953; Interscience, New York, 1954).Google Scholar
  48. 48.
    H. Poincaré, “Sur les équations aux dérivées partielles de la physique mathématique,” Am. J. Math. 12, 211–294 (1890).CrossRefMATHGoogle Scholar
  49. 49.
    H. Poincaré, “La méthode de Neumann et le problème de Dirichlet,” Acta Math. 20, 59–142 (1896).CrossRefMATHGoogle Scholar
  50. 50.
    I. I. Privalov, Cauchy Integral (Saratov, 1919) [in Russian].Google Scholar
  51. 51.
    F. Rellich, “Ein Satz über mittlere Konvergenz,” Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl., 30–35 (1930).Google Scholar
  52. 52.
    F. Riesz, “Über die Randwerte einer analytischen Funktion,” Math. Z. 18, 87–95 (1923).MathSciNetCrossRefMATHGoogle Scholar
  53. 53.
    G. Robin, “Distribution de l’électricité sur une surface fermée convexe,” C. R. Acad. Sci. Paris 104, 1834–1836 (1887).MATHGoogle Scholar
  54. 54.
    J. Schauder, “Über lineare elliptische Differentialgleichungen zweiter Ordnung,” Math. Z. 38, 257–282 (1934).MathSciNetCrossRefGoogle Scholar
  55. 55.
    J. Schauder, “Numerische Abschätzungen in elliptischen linearen Differentialgleichungen,” Stud. Math. 5, 34–42 (1934).Google Scholar
  56. 56.
    R. T. Seeley, “Singular integrals and boundary value problems,” Am. J. Math. 88, 781–809 (1966).MathSciNetCrossRefMATHGoogle Scholar
  57. 57.
    S. L. Sobolev, “Sur quelques evaluations concernant les familles des fonctions ayant des derivees a carre integrable,” C. R. Acad. Sci. URSS 1(7), 279–282 (1936).Google Scholar
  58. 58.
    S. L. Sobolev, “Méthode nouvelle à résoudre le problème de Cauchy pour les équations linéaires hyperboliques normales,” Mat. Sb. 1(1), 39–72 (1936).MATHGoogle Scholar
  59. 59.
    S. L. Sobolev, “On a theorem of functional analysis,” Mat. Sb. 4(3), 471–497 (1938) [Am. Math. Soc. Transl., Ser. 2, 34, 39–68 (1963)].MATHGoogle Scholar
  60. 60.
    S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics (Sib. Otd. Akad. Nauk SSSR, Novosibirsk, 1962; Am. Math. Soc., Providence, RI, 1991).Google Scholar
  61. 61.
    V. A. Steklov, “On the problem of existence of a function of coordinates, finite and continuous inside a given domain, that satisfies the Laplace equation for given values of its normal derivative on the surface bounding the domain,” Soobshch. Kharkov. Mat. Obshch., Ser. 2, 5(5–6), 255–286 (1897).Google Scholar
  62. 62.
    W. Stekloff, “Sur les problèmes fondamentaux de la physique mathématique,” C. R. Acad. Sci. 128, 588–591 (1899).MATHGoogle Scholar
  63. 63.
    W. Stekloff, “Les méthodes générales pour résoudre les problèmes fondamentaux de la physique mathématique,” Ann. Fac. Sci. Toulouse, Sér. 2, 2, 207–272 (1900).MathSciNetCrossRefMATHGoogle Scholar
  64. 64.
    W. Stekloff, “Mémoire sur les fonctions harmoniques de M. H. Poincaré,” Ann. Fac. Sci. Toulouse, Sér. 2, 2, 273–303 (1900).MathSciNetCrossRefGoogle Scholar
  65. 65.
    W. Stekloff, “Théorie générale des fonctions fondamentales,” Ann. Fac. Sci. Toulouse, Sér. 2, 6, 351–475 (1904).MathSciNetCrossRefMATHGoogle Scholar
  66. 66.
    W. Stekloff, “Sur la condition de fermeture des systèmes de fonctions orthogonales,” C. R. Acad. Sci. 151, 1116–1119 (1910).Google Scholar
  67. 67.
    W. Stekloff, Sur la théorie de fermeture des systemes des fonctions orthogonales dépendant d’un nombre quelconque des variables (Imper. Akad. Nauk, St. Petersburg, 1911), Zap. Imper. Akad. Nauk Fiz.-Mat. Otd., Ser. 8, 30 (4).Google Scholar
  68. 68.
    V. A. Steklov, Fundamental Problems in Mathematical Physics (Akad. Nauk, Petrograd, 1922, 1923), Parts 1, 2.Google Scholar
  69. 69.
    V. A. Steklov, Fundamental Problems in Mathematical Physics, 2nd ed., Ed. by V. S. Vladimirov (Nauka, Moscow, 1983).Google Scholar
  70. 70.
    V. S. Vladimirov, Generalized Functions in Mathematical Physics (Nauka, Moscow, 1979) [in Russian].Google Scholar
  71. 71.
    V. S. Vladimirov, Equations of Mathematical Physics (Nauka, Moscow, 1971; M. Dekker, New York, 1971); 4th ed. (Nauka, Moscow, 1981).Google Scholar
  72. 72.
    V. S. Vladimirov and I. I. Markush, Vladimir Andreevich Steklov: Scientist and Administrator (Nauka, Moscow, 1981) [in Russian].Google Scholar
  73. 73.
    N. Wiener, “The Dirichlet problem,” J. Math. Phys. 3, 127–146 (1924).MATHGoogle Scholar
  74. 74.
    S. Zaremba, “Sur la théorie de l’équation de Laplace et les méthodes de Neumann et de Robin,” Bull. Acad. Sci. Cracovie, 171–189 (1901).Google Scholar

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© Pleiades Publishing, Ltd. 2015

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia

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