New criteria for uniform approximability by harmonic functions on compact sets in ℝ2



New uniform approximability criteria formulated in terms of logarithmic capacity are obtained for approximations by harmonic functions on compact sets in ℝ2. A relationship between these approximations and analogous approximations on compact sets in ℝ3 is established.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A. Debiard and B. Gaveau, “Potentiel fin et algèbres de fonctions analytiques. I,” J. Funct. Anal. 16 (3), 289–304 (1974).CrossRefMATHGoogle Scholar
  2. 2.
    J. Deny, “Systèmes totaux de fonctions harmoniques,” Ann. Inst. Fourier 1, 103–113 (1949).MathSciNetCrossRefGoogle Scholar
  3. 3.
    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)].MATHGoogle Scholar
  4. 4.
    N. S. Landkof, Foundations of Modern Potential Theory (Nauka, Moscow, 1966; Springer, Berlin, 1972).CrossRefMATHGoogle Scholar
  5. 5.
    P. Mattila and P. V. Paramonov, “On geometric properties of harmonic Lip1-capacity,” Pac. J. Math. 171 (2), 469–491 (1995).CrossRefMATHGoogle Scholar
  6. 6.
    M. Ya. Mazalov, “Criterion of uniform approximability by harmonic functions on compact sets in R3,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 279, 120–165 (2012) [Proc. Steklov Inst. Math. 279, 110–154 (2012)].MathSciNetGoogle Scholar
  7. 7.
    M. Ya. Mazalov, P. V. Paramonov, and K. Yu. Fedorovskiy, “Conditions for C m-approximability of functions by solutions of elliptic equations,” Usp. Mat. Nauk 67 (6), 53–100 (2012) [Russ. Math. Surv. 67, 1023–1068 (2012)].MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    P. V. Paramonov, “Some new criteria for uniform approximability of functions by rational fractions,” Mat. Sb. 186 (9), 97–112 (1995) [Sb. Math. 186, 1325–1340 (1995)].MathSciNetMATHGoogle Scholar
  9. 9.
    P. V. Paramonov and J. Verdera, “Approximation by solutions of elliptic equations on closed subsets of Euclidean space,” Math. Scand. 74, 249–259 (1994).MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    J. Verdera, “C m approximation by solutions of elliptic equations, and Calderón–Zygmund operators,” Duke Math. J. 55, 157–187 (1987).MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    J. Verdera, M. S. Mel’nikov, and P. V. Paramonov, “C 1-approximation and extension of subharmonic functions,” Mat. Sb. 192 (4), 37–58 (2001) [Sb. Math. 192, 515–535 (2001)].MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    A. G. Vitushkin, “The analytic capacity of sets in problems of approximation theory,” Usp. Mat. Nauk 22 (6), 141–199 (1967) [Russ. Math. Surv. 22 (6), 139–200 (1967)].MathSciNetMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Faculty of Mechanics and MathematicsMoscow State UniversityMoscowRussia
  2. 2.Bauman Moscow State Technical UniversityMoscowRussia

Personalised recommendations