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Sets of Harmonicity for Finely Harmonic Functions

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Abstract

Given an open set U in R n (n≥3) and a dense open subset V of U, it is shown that there is a finely harmonic function u on U such that V is the largest open subset of U on which u is harmonic. This result, which establishes the sharpness of a theorem of Fuglede, is obtained following a consideration of fine cluster sets of arbitrary functions.

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References

  1. Armitage, D.H. and Gardiner, S.J.: Classical Potential Theory, Springer, London, 2001.

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  2. Fuglede, B.: Finely Harmonic Functions, Lecture Notes in Math. 289, Springer, Berlin, 1972.

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  3. Fuglede, B.: ‘Fonctions harmoniques et fonctions finement harmoniques’, Ann. Inst. Fourier (Grenoble) 24(4) (1975), 77–91.

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  4. Gardiner, S.J.: Harmonic Approximation, London Math. Soc. Lecture Note Series 221, Cambridge Univ. Press, 1995.

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Gardiner, S.J. Sets of Harmonicity for Finely Harmonic Functions. Potential Analysis 21, 1–6 (2004). https://doi.org/10.1023/B:POTA.0000021331.19636.63

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  • DOI: https://doi.org/10.1023/B:POTA.0000021331.19636.63

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