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 Mat. Zametki, 2012, Volume 92, Issue 3, Pages 447–458 (Mi mz8750)

Integral Properties of the Classical Warping Function of a Simply Connected Domain

R. G. Salakhudinov

N. G. Chebotarev Research Institute of Mathematics and Mechanics, Kazan State University

Abstract: Let $u(z,G)$ be the classical warping function of a simply connected domain $G$. We prove that the $L^p$-norms of the warping function with different exponents are related by a sharp isoperimetric inequality, including the functional $u(G)=\sup_{x\in G}u(x,G)$. A particular case of our result is the classical Payne inequality for the torsional rigidity of a domain. On the basis of the warping function, we construct a new physical functional possessing the isoperimetric monotonicity property. For a class of integrals depending on the warping function, we also obtain a priori estimates in terms of the $L^p$-norms of the warping function as well as the functional $u(G)$. In the proof, we use the estimation technique on level lines proposed by Payne.

Keywords: warping function, isoperimetric inequality, isoperimetric monotonicity, torsional rigidity, Payne inequality, level lines, Schwartz symmetrization

DOI: https://doi.org/10.4213/mzm8750

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English version:
Mathematical Notes, 2012, 92:3, 412–421

Bibliographic databases:

UDC: 517.5+517.956.225

Citation: R. G. Salakhudinov, “Integral Properties of the Classical Warping Function of a Simply Connected Domain”, Mat. Zametki, 92:3 (2012), 447–458; Math. Notes, 92:3 (2012), 412–421

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/mz8750
• https://doi.org/10.4213/mzm8750
• http://mi.mathnet.ru/eng/mz/v92/i3/p447

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This publication is cited in the following articles:
1. R. G. Salakhudinov, “Payne type inequalities for $L^p$-norms of the warping functions”, J. Math. Anal. Appl., 410:2 (2014), 659–669
2. R. G. Salakhudinov, “Some properties of functionals on level sets”, Ufa Math. J., 11:2 (2019), 114–124
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