Universal boundary value problem for equations of mathematical physics

  • I. V. Volovich
  • V. Zh. Sakbaev


A new statement of a boundary value problem for partial differential equations is discussed. An arbitrary solution to a linear elliptic, hyperbolic, or parabolic second-order differential equation is considered in a given domain of Euclidean space without any constraints imposed on the boundary values of the solution or its derivatives. The following question is studied: What conditions should hold for the boundary values of a function and its normal derivative if this function is a solution to the linear differential equation under consideration? A linear integral equation is defined for the boundary values of a solution and its normal derivative; this equation is called a universal boundary value equation. A universal boundary value problem is a linear differential equation together with a universal boundary value equation. In this paper, the universal boundary value problem is studied for equations of mathematical physics such as the Laplace equation, wave equation, and heat equation. Applications of the analysis of the universal boundary value problem to problems of cosmology and quantum mechanics are pointed out.


Wave Equation STEKLOV Institute Fundamental Solution Heat Equation Laplace Equation 
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  1. 1.
    J. Hadamard, Le problème de Cauchy et les équations aux dérivées partielles linéaires hyperboliques (Hermann, Paris, 1932; Nauka, Moscow, 1978).Google Scholar
  2. 2.
    V. I. Arnol’d, “Small denominators. I: Mappings of the circumference onto itself,” Izv. Akad. Nauk SSSR, Ser. Mat. 25(1), 21–86 (1961) [Am. Math. Soc. Transl., Ser. 2, 46, 213–284 (1965)].MathSciNetGoogle Scholar
  3. 3.
    A. V. Bitsadze, Some Classes of Partial Differential Equations (Nauka, Moscow, 1981; Gordon & Breach, New York, 1988).MATHGoogle Scholar
  4. 4.
    V. P. Burskii, Methods of Analysis of Boundary Value Problems for General Differential Equations (Naukova Dumka, Kiev, 2002) [in Russian].Google Scholar
  5. 5.
    A. D. Ventsel’ and M. I. Freidlin, Fluctuations in Dynamical Systems under Small Random Perturbations (Nauka, Moscow, 1979); Engl. transl.: M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems (Springer, New York, 1984).Google Scholar
  6. 6.
    M. I. Vishik, “On general boundary problems for elliptic differential equations,” Tr. Mosk. Mat. Obshch. 1, 187–246 (1952) [Am. Math. Soc. Transl., Ser. 2, 24, 107–172 (1963)].MATHGoogle Scholar
  7. 7.
    M. I. Vishik and S. L. Sobolev, “General statement of boundary value problems for elliptic partial differential equations,” Dokl. Akad. Nauk SSSR 111(3), 521–523 (1956).MATHMathSciNetGoogle Scholar
  8. 8.
    V. S. Vladimirov, Equations of Mathematical Physics (Nauka, Moscow, 1981), 4th ed.; Engl transl. of the 1st ed.: Equations of Mathematical Physics (M. Dekker, New York, 1971).Google Scholar
  9. 9.
    V. S. Vladimirov, Generalized Functions in Mathematical Physics (Nauka, Moscow, 1976; Mir, Moscow, 1979).Google Scholar
  10. 10.
    V. S. Vladimirov, “Solutions of p-adic string equations,” Teor. Mat. Fiz. 167(2), 163–170 (2011) [Theor. Math. Phys. 167, 539–546 (2011)].CrossRefGoogle Scholar
  11. 11.
    V. S. Vladimirov and I. V. Volovich, “Local and nonlocal currents for nonlinear equations,” Teor. Mat. Fiz. 62(1), 3–29 (1985) [Theor. Math. Phys. 62, 1–20 (1985)].CrossRefMathSciNetGoogle Scholar
  12. 12.
    I. V. Volovich, O. V. Groshev, N. A. Gusev, and E. A. Kuryanovich, “On solutions to the wave equation on a non-globally hyperbolic manifold,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 265, 273–287 (2009) [Proc. Steklov Inst. Math. 265, 262–275 (2009)]; arXiv: 0903.0741 [math-ph].MathSciNetGoogle Scholar
  13. 13.
    E. B. Dynkin, Markov Processes (Fizmatgiz, Moscow, 1963; Springer, Berlin, 1965).MATHGoogle Scholar
  14. 14.
    N. V. Krylov, “On the solutions to second-order elliptic equations,” Usp. Mat. Nauk 21(2), 233–235 (1966).Google Scholar
  15. 15.
    C. Miranda, Partial Differential Equations of Elliptic Type (Springer, Berlin, 1970; Inostrannaya Literatura, Moscow, 1957).CrossRefMATHGoogle Scholar
  16. 16.
    S. G. Mikhlin, Linear Partial Differential Equations (Vysshaya Shkola, Moscow, 1977) [in Russian].Google Scholar
  17. 17.
    O. A. Oleinik and E. V. Radkevich, Equations with Nonnegative Characteristic Form (Mosk. Gos. Univ., Moscow, 2010); partial Engl. transl.: O. A. Oleĭnik and E. V. Radkevič, Second Order Equations with Nonnegative Characteristic Form (Plenum, New York, 1973).Google Scholar
  18. 18.
    S. L. Sobolev, “An example of a well-posed boundary value problem for the equation of string vibrations with data on the whole boundary,” Dokl. Akad. Nauk SSSR 109(4), 707–709 (1956).MATHMathSciNetGoogle Scholar
  19. 19.
    V. Zh. Sakbaev, “Spectral aspects of regularization of the Cauchy problem for a degenerate equation,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 261, 258–267 (2008) [Proc. Steklov Inst. Math. 261, 253–261 (2008)].MathSciNetGoogle Scholar
  20. 20.
    V. Zh. Sakbaev, Cauchy Problem for a Degenerate Linear Differential Equation and Averaging Its Approximating Regularizations (Ross. Univ. Druzhby Narodov, Moscow, 2012), Sovrem. Mat., Fundam. Napr. 43.Google Scholar
  21. 21.
    A. N. Tikhonov and A. A. Samarskii, Equations of Mathematical Physics, 6th ed. (Mosk. Gos. Univ., Moscow, 1999); Engl. transl. of the 2nd ed.: Equations of Mathematical Physics (Pergamon, Oxford, 1963).Google Scholar
  22. 22.
    L. Hörmander, “On the theory of general partial differential operators,” Acta Math. 94, 161–248 (1955).CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    I. Ya. Aref’eva and I. V. Volovich, “Cosmological daemon,” J. High Energy Phys., No. 8, 102 (2011); arXiv: 1103.0273 [hep-th].Google Scholar
  24. 24.
    G. Fichera, “On a unified theory of boundary value problems for elliptic-parabolic equations of second order,” in Boundary Problems in Differential Equations (Univ. Wisconsin Press, Madison, 1960), pp. 97–120.Google Scholar
  25. 25.
    M. Ohya and I. Volovich, Mathematical Foundations of Quantum Information and Computation and Its Applications to Nano- and Bio-systems (Springer, Dordrecht, 2011).CrossRefMATHGoogle Scholar
  26. 26.
    V. Zh. Sakbaev and I. V. Volovich, “A universal boundary value problem for partial differential equations,” arXiv: 1312.4302 [math.AP].Google Scholar

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© Pleiades Publishing, Ltd. 2014

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia
  2. 2.Moscow Institute of Physics and Technology (State University)Dolgoprudnyi, Moscow oblastRussia

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