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Harmonic Functions on Manifolds
The Annals of Mathematics, 1997For an open manifold \(M^n\), given a point \(p\in M^n\), let \(r\) be the distance from \(p\). Define \({\mathcal H}_d(M^n)\) to be the linear space of harmonic functions with order of growth at most \(d\). The main result of this paper is a proof of the following Yau's conjecture: Conjecture. For an open manifold with nonnegative Ricci curvature, the
William P. Minicozzi, Tobias H. Colding
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Sets of Harmonicity for Finely Harmonic Functions
Potential Analysis, 2004The author establishes the sharpness of a theorem of Fuglede. \textit{B. Fuglede} [Ann. Inst. Fourier 24, No. 4, 77--91 (1974; Zbl 0287.31003)] observed the following result. Let \(U\) be an open set in \({\mathbb R}^n\) (\(n\geq 2\)). If \(u\) is finely harmonic on \(U\), then there is a dense open subset \(V\) of \(U\) on which \(u\) is harmonic. The
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On positive harmonic functions
Mathematical Proceedings of the Cambridge Philosophical Society, 1952Any harmonic function which is defined and positive in the half-plane η > 0 may be expressed bywhere C is a non-negative number, and G(x) is a bounded non-decreasing function. For a simple proof see Loomis and Widder (2). Let us writewhere w(z) is a regular function of z in η > 0, and satisfies the following conditions: (i) w(z) is real and ...
A. C. Allen, R. A. Rankin
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On discrete harmonic functions
Mathematical Proceedings of the Cambridge Philosophical Society, 1949A function f(x1, x2) of two real variables x1, x2 which are restricted to rational integers will be called discrete harmonic (d.h.) if it satisfies the difference equationThis equation can be considered as the direct analogue either of the differential equationor of the integral equationin the notation normally employed to harmonic functions.
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On a Property of Harmonic Functions
Zeitschrift für Analysis und ihre Anwendungen, 1995If we divide the space \mathbb R^n into two disjoint areas with one common hypersurface and define a harmonic function in each part of these areas such that their gradients vanish at infinity and the normal components of their gradients are equal on the hypersurface, then for some ...
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On the approximation of harmonic functions by harmonic polynomials
Siberian Mathematical Journal, 1974openaire +2 more sources
On separation by harmonic functions [PDF]
Acta Mathematica Academiae Scientiarum Hungaricae, 1964 openaire +2 more sources