Integral properties of the classical warping function of a simply connected domain |
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Authors: | R G Salakhudinov |
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Institution: | 1. Lobachevsky Research Institute of Mathematics and Mechanics, Kazan Federal University, Kazan, Russia
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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∈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. |
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