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Erschienen in:
2009 | OriginalPaper | Buchkapitel
Multidimensional Harmonic Functions Analogues of Sharp Real-part Theorems in Complex Function Theory
verfasst von : Gershon Kresin
Erschienen in: Analysis, Partial Differential Equations and Applications
Verlag: Birkhäuser Basel
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In the present paper, the sharp multidimensional analogues of Lindelöf inequality and similar estimates for analytic functions are considered. Using a sharp inequality for the gradient of a bounded or semibounded harmonic function in a ball, one arrives at improved estimates (compared with the known ones) for the gradient of harmonic functions in an arbitrary subdomain of ℝ
n
. A representation of the sharp constant in a pointwise estimate of the gradient of a harmonic function in a half-space is obtained under the assumption that function’s boundary values belong to
L
p
. This representation is realized in the three-dimensional case and the values of sharp constants are explicitly given for
p
=1, 2,∞.