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Sharp integral inequalities for harmonic functions. (English) Zbl 1173.26321
Motivated by Carleman’s proof of the isoperimetric inequality in the plane, the authors study some sharp integral inequalities for harmonic functions on the upper half-space. They derive the regularity for nonnegative solutions of the associated integral system and some Liouville-type theorems. One of the key ingredients in the proofs is a symmetrization method due to E. H. Lieb, based on the Riesz rearrangement inequalities and its strong form [Studies Appl. Math. 57, 93–105 (1977; Zbl 0369.35022)]. The results and methods are, however, too complicated to be stated here.

MSC:
26D15 Inequalities for sums, series and integrals
31B10 Integral representations, integral operators, integral equations methods in higher dimensions
46G15 Functional analytic lifting theory
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