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Nodal sets of harmonic functions. (English) Zbl 1141.31002

In the present paper, the author discusses the relation between the growth of harmonic functions and the growth of nodal sets of those functions. The growth of harmonic functions is measured by their frequency. For any harmonic function \(u\) in unit ball \(B_1\subset \mathbb{R}^n\), the frequency is defined as \[ N= {\int_{B_1}|\nabla u|^2\over \int_{\partial B_1} u^2}. \] The frequency controls the growth of the harmonic functions. In the present paper, the author discussed how the frequency controls the size of nodal sets.

MSC:

31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
31-02 Research exposition (monographs, survey articles) pertaining to potential theory
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