Approximation by Solutions of Elliptic Equations and Extension of Subharmonic Functions

  • Paul Gauthier
  • Petr V. Paramonov
Part of the Fields Institute Communications book series (FIC, volume 81)


In this review we present the main results jointly obtained by the authors and André Boivin (1955–2014) during the last 20 years. We also recall some important theorems obtained with colleagues and give new applications of the above mentioned results. Several open problems are also formulated.


Elliptic operator Banach space of distributions Approximation on closed sets L-analytic and L-meromorphic functions Localization operator Cm-extension of subharmonic functions 

2010 Mathematics Subject Classification.

Primary 30E10 31A05; Secondary 31A35 31B05 35B60 35J30 



Paul Gauthier was supported by NSERC (Canada) and Entente France-Québec. Petr V. Paramonov was partially supported by the Programme for the Support of Leading Scientific Schools of the Russian Federation (grant NSh-9110.2016.1).


  1. 1.
    S. Agmon, Lectures on Elliptic Boundary Value Problems, D.Van Nostrand, Princeton - Toronto - New York - London, 1965.Google Scholar
  2. 2.
    N. U. Arakelyan, Uniform approximation on closed sets by entire functions (Russian), Izv. AN SSSR, Ser. Mat., 28 (1964), 1187–1206.Google Scholar
  3. 3.
    A. Boivin P. M. Gauthier and P. V. Paramonov, Approximation on closed sets by analytic or meromorphic solutions of elliptic equations and applications, Canad. J. Math. 54:5 (2002), 945–969.Google Scholar
  4. 4.
    A. Boivin, P. M. Gauthier and P. V. Paramonov, On uniform approximation by n-analytic functions on closed sets in C, Izvestiya: Mathematics. 68:3 (2004), 447–459.Google Scholar
  5. 5.
    A. Boivin, P.M. Gauthier, P.V. Paramonov, C m -subharmonic extension of Runge type from closed to open subsets of R N, Proc. Steklov Math. Inst., 279 (2012), 207–214.Google Scholar
  6. 6.
    A. Boivin, P. M. Gauthier, P. V. Paramonov, Runge- and Walsh-type extensions of subharmonic functions on open Riemann surfaces, Sbornik: Mathematics. 206:1 (2015), 3–23.Google Scholar
  7. 7.
    A. Boivin and P. V. Paramonov, Approximation by meromorphic and entire solutions of elliptic equations in Banach spaces of distributions, Sbornik: Mathematics. 189:4 (1998), 481–502.Google Scholar
  8. 8.
    A. Boivin and P. V. Paramonov, On radial limit functions for entire solutions of second order elliptic equations in R 2, Publ. Mat. 42:2 (1998), 509–519.Google Scholar
  9. 9.
    J. J. Carmona, P. V. Paramonov and K. Yu. Fedorovskiy, On uniform approximation by polyanalytic polynomials and the Dirichlet problem for bianalytic functions. Sbornik: Mathematics. 193:9–10 (2002), 1469–1492.Google Scholar
  10. 10.
    A. Dufresnoy, P. M. Gauthier and W. H. Ow, Uniform approximation on closed sets by solutions of elliptic partial differential equations, Complex Variables. 6 (1986), 235–247.Google Scholar
  11. 11.
    D. Gaier, Lectures on Complex Approximation, Birkhäuser, Boston - Basel - Stuttgart, 1987.Google Scholar
  12. 12.
    S. J. Gardiner, Harmonic Approximation, London Mathematical Society Lecture Notes Series 221, Cambridge University Press, 1995.Google Scholar
  13. 13.
    P. M. Gauthier, Subharmonic extension and approximation, Canadian Math. Bull. 37 (1994), 46–53.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    P. M. Gauthier and P. V. Paramonov, Approximation by harmonic functions in C 1 -norm, and the harmonic C 1 -content of compact sets in R n, Math. Notes 53 (1993), no. 3–4,Google Scholar
  15. 15.
    Yu.A. Gorokhov, Approximation by harmonic functions in the C m -norm and harmonic C m -capacity of compact sets in R n, Mathem. Notes 62:3 (1997), 314–322.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    R. C. Gunning, R. Narasimhan, Immersion of open Riemann surfaces, Math. Ann. 174 (1967), 103–108.MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    L. Hörmander, The Analysis of Linear Partial Differential Operators I, Springer-Verlag, Berlin New York, 1983.zbMATHGoogle Scholar
  18. 18.
    J. Král, Some extension results concerning harmonic functions, J. London. Math. Soc. (2). 28:1 (1983), 62–70.Google Scholar
  19. 19.
    M. Ya. Mazalov, A criterion for uniform approximability on arbitrary compact set by solutions of elliptic equations. Sbornik: Mathematics. 199:1–2 (2008), 13–44.Google Scholar
  20. 20.
    M. Ya. Mazalov, P. V. Paramonov, K. Yu. Fedorovskii, Conditions for C m -approximability of functions by solutions of elliptic equations. Russian Math. Surveys 67:6 (2012), 1023–1068.Google Scholar
  21. 21.
    M. Ya. Mazalov, P. V. Paramonov, Criteria for C m -approximability by bianalytic functions on planar compact sets. Sbornik: Mathematics. 206:2 (2015), 242–281.Google Scholar
  22. 22.
    M. S. Mel’nikov, P.V. Paramonov and J. Verdera, C 1 -approximation and extension of subharmonic functions. Sbornik: Mathematics. 192:4 (2001), 515–535.MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    M. S. Mel’nikov, P.V. Paramonov. C 1 -extension of subharmonic functions from closed Jordan domains in R 2. Izvestiya: Mathematics. 68:6 (2004), 1165–1178.MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    A. A. Nersesjan, Uniform and tangential approximation by meromorphic functions (Russian), Izv. AN Arm. SSR, Ser. Matem.7 (1972), 405–412.Google Scholar
  25. 25.
    A. G. O’Farrell, T-invariance, Proc. Roy. Irish Acad. 92A:2 (1992), 185–203.zbMATHGoogle Scholar
  26. 26.
    P. V. Paramonov. C 1 -extension and C 1 -reflection of subharmonic functions from Lyapunov-Dini domains into R N. Sbornik: Mathematics. 199:12 (2008), 1809–1846.MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    P. V. Paramonov. On C m -extension of subharmonic functions from Lyapunov-Dini domains to R N. Math. Notes 89:1 (2011), 160–164.MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    P. V. Paramonov, On C m -subharmonic extension of Walsh-type, Centre de Recherches Mathématiques. CRM Proceedings and Lecture Notes, 55 (2012), 201–209.MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    P. V. Paramonov and J. Verdera, Approximation by solutions of elliptic equations on closed subsets of Euclidean space, Math. Scand. 74 (1994), 249–259.MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    A. Phillips, Submersions of open manifolds, Topology 6 (1967), 171–206.MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    T. Radó, Über eine nicht fortsetzbare Riemannsche Mannigflatigkeit, Math. Z., 20 (1924), 1–6.MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    A. Roth, Approximationseigenschaften und Strahlengrenzwerte meromorpher und ganzer Funktionen, Comment. Math. Helv. 11 (1938), 77–125.MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    W. Rudin, Real and Complex Analysis, Third Edition, McGraw Hill, New York & als, 1987.zbMATHGoogle Scholar
  34. 34.
    C. Runge, Zur theorie der eindeutigen analytischen Funktionen. Acta Math. 6 (1885), 228–244.Google Scholar
  35. 35.
    E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, New Jersey, 1970.Google Scholar
  36. 36.
    J. Verdera, C m approximation by solutions of elliptic equations, and Calderón-Zygmund operators, Duke Math. J. 55 (1987), 157–187.MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    A. G. Vitushkin, Analytic capacity of sets in problems of approximation theory, Russian Math. Surveys (English Translation) 22 (1967), 139–200.MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    A.M. Vorontsov, Estimates of C m -capacity of compact sets in R N, Mathem. Notes 75:6 (2004), 751–764.MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    K.-O. Widman. Inequalities for the Green function and boundary continuity of the gradient of solutions of elliptic differential equations. Math. Scand. 21 (1967), 17–37.MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    A. B. Zaitsev, Uniform approximation by polynomial solutions of second-order elliptic equations, and the corresponding Dirichlet problem, Proc. Steklov Math. Inst. 253:1 (2006), 57–70.CrossRefzbMATHGoogle Scholar
  41. 41.
    O. A. Zorina, C m -extension of subholomorphic functions from plane Jordan domains, Izvestiya: Math. 69:6 (2005), 1099–1111.MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Département de Mathématiques et de StatistiqueUniversité de MontréalMontréalCanada
  2. 2.Mechanics and Mathematics FacultyMoscow State (Lomonosov) UniversityMoscowRussia

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