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
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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