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Harmonic functions vanishing on a cone

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Abstract

Let Z be a quadratic harmonic cone in ℝ3. We consider the family \(\mathcal{H}(Z)\) of all harmonic functions vanishing on Z. Is \(\mathcal{H}(Z)\) finite or infinite dimensional? Some aspects of this question go back to as early as the 19th century. To the best of our knowledge, no nondegenerate quadratic harmonic cone exists for which the answer to this question is known. In this paper we study the right circular harmonic cone and give evidence that the family of harmonic functions vanishing on it is, maybe surprisingly, finite dimensional. We introduce an arithmetic method to handle this question which extends ideas of Holt and Ille and is reminiscent of Hensel’s Lemma.

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Authors and Affiliations

  1. Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Givat Ram, Jerusalem, 9190401, Israel

    Dan Mangoubi & Adi Weller Weiser

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  1. Dan Mangoubi
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  2. Adi Weller Weiser
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Correspondence to Adi Weller Weiser.

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Mangoubi, D., Weller Weiser, A. Harmonic functions vanishing on a cone. Isr. J. Math. 230, 563–581 (2019). https://doi.org/10.1007/s11856-018-1823-9

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  • DOI: https://doi.org/10.1007/s11856-018-1823-9

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