Superharmonic extension and harmonic approximation

Stephen J. Gardiner

Annales de l'institut Fourier (1994)

  • Volume: 44, Issue: 1, page 65-91
  • ISSN: 0373-0956

Abstract

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Let Ω be an open set in ℝ n and E be a subset of Ω . We characterize those pairs ( Ω , E ) which permit the extension of superharmonic functions from E to Ω , or the approximation of functions on E by harmonic functions on Ω .

How to cite

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Gardiner, Stephen J.. "Superharmonic extension and harmonic approximation." Annales de l'institut Fourier 44.1 (1994): 65-91. <http://eudml.org/doc/75061>.

@article{Gardiner1994,
abstract = {Let $\Omega $ be an open set in $\{\Bbb R\}^n$ and $E$ be a subset of $\Omega $. We characterize those pairs $(\Omega , E)$ which permit the extension of superharmonic functions from $E$ to $\Omega $, or the approximation of functions on $E$ by harmonic functions on $\Omega $.},
author = {Gardiner, Stephen J.},
journal = {Annales de l'institut Fourier},
keywords = {harmonic measure; extension theorem; thin set; superharmonic function},
language = {eng},
number = {1},
pages = {65-91},
publisher = {Association des Annales de l'Institut Fourier},
title = {Superharmonic extension and harmonic approximation},
url = {http://eudml.org/doc/75061},
volume = {44},
year = {1994},
}

TY - JOUR
AU - Gardiner, Stephen J.
TI - Superharmonic extension and harmonic approximation
JO - Annales de l'institut Fourier
PY - 1994
PB - Association des Annales de l'Institut Fourier
VL - 44
IS - 1
SP - 65
EP - 91
AB - Let $\Omega $ be an open set in ${\Bbb R}^n$ and $E$ be a subset of $\Omega $. We characterize those pairs $(\Omega , E)$ which permit the extension of superharmonic functions from $E$ to $\Omega $, or the approximation of functions on $E$ by harmonic functions on $\Omega $.
LA - eng
KW - harmonic measure; extension theorem; thin set; superharmonic function
UR - http://eudml.org/doc/75061
ER -

References

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  1. [1] N.U. ARAKELJAN, Uniform and tangential approximations by analytic functions, Izv. Akad. Nauk Armjan. SSR, Ser. Mat., 3 (1968), 273-286 (Russian); Amer. Math. Soc. Transl., (2) 122 (1984), 85-97. Zbl0552.30028MR43 #530
  2. [2] D.H. ARMITAGE, On the extension of superharmonic functions, J. London Math. Soc., (2) 4 (1971), 215-230. Zbl0223.31009MR45 #8868
  3. [3] D.H. ARMITAGE and M. GOLDSTEIN, Tangential harmonic approximation on relatively closed sets, Proc. London Math. Soc. (3) 68 (1994), 112-126. Zbl0795.31002MR94i:31005
  4. [4] T. BAGBY and P. BLANCHET, Uniform harmonic approximation on Riemannian manifolds, J. Analyse Math., to appear. Zbl0806.31004
  5. [5] T. BAGBY and P.M. GAUTHIER, Uniform approximation by global harmonic functions, in Approximation by Solutions of Partial Differential Equations, ed. B. Fuglede et al., NATO ASI Series, Kluwer, Dordrecht, 1992, pp. 15-26. Zbl0759.41017MR93g:31015
  6. [6] D.M. CAMPBELL, J.G. CLUNIE and W.K. HAYMAN, Research problems in complex analysis, in Aspects of Contemporary Complex Analysis, ed. D.A. Brannan and J.G. Clunie, Academic Press, New York, 1980, pp. 527-572. Zbl0497.30001MR83b:30002
  7. [7] J. DENY, Systèmes totaux de fonctions harmoniques, Ann. Inst. Fourier, Grenoble, 1 (1949), 103-113. 
  8. [8] J.L. DOOB, Semimartingales and subharmonic functions, Trans. Amer. Math. Soc., 77 (1954), 86-121. Zbl0059.12205MR16,269a
  9. [9] J.L. DOOB, Classical potential theory and its probabilistic counterpart, Springer, New York, 1983. Zbl0549.31001
  10. [10] P.M. GAUTHIER, M. GOLDSTEIN and W.H. OW, Uniform approximation on unbounded sets by harmonic functions with logarithmic singularities, Trans. Amer. Math. Soc., 261 (1980), 169-183. Zbl0447.30035MR81m:30033
  11. [11] P.M. GAUTHIER, M. GOLDSTEIN and W.H. OW, Uniform approximation on closed sets by harmonic functions with Newtonian singularities, J. London Math. Soc., (2) 28 (1983), 71-82. Zbl0525.31002MR84j:31009
  12. [12] M. GOLDSTEIN and W.H. OW, A characterization of harmonic Arakelyan sets, Proc. Amer. Math. Soc., 119 (1993), 811-816. Zbl0787.31003MR93m:31005
  13. [13] L.L. HELMS, Introduction to potential theory, Krieger, New York, 1975. Zbl0188.17203
  14. [14] M.V. KELDYŠ, On the solvability and stability of the Dirichlet problem, Uspehi Mat. Nauk, 8 (1941), 171-231 (Russian); Amer. Math. Soc. Transl., 51 (1966), 1-73. Zbl0179.43901JFM67.0340.02
  15. [15] M. LABRÈCHE, De l'approximation harmonique uniforme, Doctoral thesis, Université de Montréal, 1982. 
  16. [16] PREMALATHA, On a superharmonic extension, Rev. Roum. Math. Pures Appl., 26 (1981), 631-640. Zbl0468.31004MR82i:31013
  17. [17] A.A. SAGINJAN, On tangential harmonic approximation and some related problems, Complex Analysis I, Lecture Notes in Mathematics 1275, Springer, Berlin (1987), 280-286. Zbl0647.31002MR89g:31005

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