Geodesic
Integral Properties of the Classical Warping Function of a Simply Connected Domain
Matematičeskie zametki, Tome 92 (2012) no. 3, pp. 447-458.

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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. 
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R. G. Salakhudinov. Integral Properties of the Classical Warping Function of a Simply Connected Domain. Matematičeskie zametki, Tome 92 (2012) no. 3, pp. 447-458. http://geodesic.mathdoc.fr/item/MZM_2012_92_3_a12/

[1] G. Polia, G. Sege, Izoperimetricheskie neravenstva v matematicheskoi fizike, Fizmatlit, M., 1962 | MR | Zbl

[2] B. Sen-Venan, Memuar o kruchenii prizm. Memuar ob izgibe prizm, GIFML, M., 1961 | Zbl

[3] L. E. Payne, “Some isoperimetric inequalities in the torsion problem for multiply connected regions”, Studies in Mathematical analysis and Related Topics, Stanford Univ. Press, Stanford, CA, 1962, 270–280 | MR | Zbl

[4] F. G. Avkhadiev, “Reshenie obobschennoi zadachi Sen-Venana”, Matem. sb., 189:12 (1998), 3–12 | MR | Zbl

[5] R. G. Salahudinov, “Isoperimetric inequality for torsional rigidity in the complex plane”, J. Inequal. Appl., 6:3 (2001), 253–260 | MR | Zbl

[6] R. G. Salakhudinov, “Dvustoronnie otsenki $L^p$-norm funktsii napryazheniya vypuklykh oblastei v $\mathbb R^n$”, Izv. vuzov. Matem., 2006, no. 3, 41–49 | MR | Zbl

[7] R. Bañuelos, M. van den Berg, T. Carroll, “Torsional rigidity and expected lifetime of Brownian motion”, J. London Math. Soc. (2), 66:2 (2002), 499–512 | DOI | MR | Zbl

[8] R. Bañuelos, T. Carroll, “Brownian motion and fundamental frequency of a drum”, Duke Math. J., 75:3 (1994), 575–602 | DOI | MR | Zbl

[9] C. Bandle, Isoperimetric Inequalities and Applications, Monogr. Stud. Math., 7, Pitman, Boston, MA, 1980 | MR | Zbl

[10] J. Hersch, “Isoperimetric monotonicity: some properties and conjectures (connections between isoperimetric inequalities)”, SIAM Rev., 30:4 (1988), 551–577 | DOI | MR | Zbl

[11] M.-Th. Kohler-Jobin, “Une propriété de monotonie isopérimétrique qui contient plusieurs théorèmes classiques”, C. R. Acad. Sci. Paris Sér. A-B, 284 (1977), 917–920 | MR | Zbl

[12] R. G. Salahudinov, “Isoperimetric inequalities for $l^p$-norms of the distance function to the boundary”, Uchën. zap. Kazan. gos. un-ta. Ser. Fiz.-matem. nauki, 148, kn. 2, Izd-vo Kazanskogo un-ta, Kazan, 2006, 151–162 | Zbl

[13] R. G. Salakhudinov, “Izoperimetricheskaya monotonnost $L^p$-normy funktsii napryazheniya ploskoi odnosvyaznoi oblasti”, Izv. vuzov. Matem., 2010, no. 8, 59–68 | MR | Zbl