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Kazanskii Gosudarstvennyi Universitet. Uchenye Zapiski. Seriya Fiziko-Matematichaskie Nauki, 2006, Volume 148, Book 2, Pages 151–162
(Mi uzku554)
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This article is cited in 4 scientific papers (total in 4 papers)
Isoperimetric inequalities for $l^p$-norms of the distance function to the boundary
R. G. Salahudinov N. G. Chebotarev Research Institute of Mathematics and Mechanics, Kazan State University
Abstract:
The main goal of the paper is to prove that $L^p$-norms of $dist(x,\partial G)$ and $dist^{-1}(x,\partial G)$ are decreasing functions of $p$, where $G$ is a domain in ${\mathbb R}^n(n\ge2)$. We also obtain a sharp estimation of the rate of decreasing for these norms using $L^p$ —norms of the distance function for a consistent ball. We prove a new isoperimetric inequality for $L^p$ —norms of $dist(x,\partial G)$, this inequality is analogous to the inequality of $L^p$–norms of the conformal radii (see Avkhadiev F.G., Salahudinov R.G. // J. of Inequal. & \rm Appl. – 2002. – V. 7, No 4. – P. 593–601).
Note that $L^2$-norm of $dist(x,\partial G)$ plays an important role to investigate the torsional rigidity in Mathematical Physics (see, for instance, Avkhadiev F.G. // Sbornik: Math. – 1998. – V. 189, No 12. – P. 1739–1748; Banuelos R., van den Berg M., Carroll T. // J. London Math. Soc. – 2002. – V. 66, No 2. – P. 499–512). As a consequence we get new inequalities in the torsional rigidity problem.
Also we generalize the $n$-dimensional isoperimetric inequality.
Received: 21.03.2006
Citation:
R. G. Salahudinov, “Isoperimetric inequalities for $l^p$-norms of the distance function to the boundary”, Kazan. Gos. Univ. Uchen. Zap. Ser. Fiz.-Mat. Nauki, 148, no. 2, Kazan University, Kazan, 2006, 151–162
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