Some properties of functionals on level sets

R. G. Salakhudinov

Ufa mathematical journal, Tome 11 (2019) no. 2, pp. 114-124

Résumé

In the paper we consider special functionals on a planar domain $G$ constructed by means of the distance to the boundary $\partial G$ and a classical warping function. The functionals depending on the distance function are considered for simply-connected domains. We also study the functionals depending on the warping function for a finite-connected domain. We prove that the property of isoperimetric monotonicity with respect to a free parameter gives rise to another monotonicity, namely, the monotonicity of the functionals considered as the functions of the sets defined on subsets of the domain. Some partial cases of the inequality were earlier obtained by Payne. We note that the inequalities were successfully applied for justifying new estimates for the torsional rigidity of simply-connected and multiply-connected domains. In particular, we construct new functionals of domains monotone in both its variables. Moreover, we find sharp estimates of variation rate of the functions, that is, we obtain sharp estimates of their derivatives.

Détail

Keywords: distance to boundary, warping function, Payne type inequality, isoperimetric inequality, isoperimetric monotonicity.

Bibliographie

[1] R.G. Salakhudinov, “Integral properties of the classical warping function of a simply connected domain”, Math. Notes, 92:3 (2012), 412–421 | DOI | DOI | MR | Zbl

[2] F.G. Avkhadiev, “Solution of the generalized Saint Venant problem”, Sb. Math., 189:12 (1998), 1739–1748 | DOI | DOI | MR | Zbl

[3] F.G. Avkhadiev, “Hardy type inequalities in higher dimensions with explicit estimate of constants”, Lobachevskii J. Math., 21 (2006), 3–31 | MR | Zbl

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

[5] R.G. Salakhudinov, “Two-sided estimates for the Lp-norms of the stress function for convex domains in $\mathbb{R}^n$”, Russ. Math. Iz. VUZ, 50:3 (2006), 39–46 | MR | Zbl

[6] L.E. Payne, “Some inequalities in the torsion problem for multiply connected regions”, Studies in Mathematical analysis and Related Topics, Essays in honor of G. Pólya, Stanford University Press, Stanford, California, 1962, 270–280 | MR

[7] G. Pólya, G. Szegö, Isoperimetric inequalities in mathematical physics, Ann. Math. Stud., 27, Princeton Univ. Press, Princeton, 1951 | MR | Zbl

[8] C. Bandle, Isoperimetric inequalities and applications, Pitman Advanced Publishing Program, Boston–London–Melbourne, 1980 | MR | Zbl

[9] R.G. Salakhudinov, “Refined inequalities for euclidian moments of a domain with respect to its boundary”, SIAM J. Math. Anal., 44:4 (2012), 2949–2961 | DOI | MR | Zbl

[10] R.G. Salakhudinov, “Isoperimetric properties of Euclidean boundary moments of a simply connected domain”, Russ. Math. Iz. VUZ, 57:8 (2013), 57–69 | DOI | MR | Zbl

[11] R.G. Salakhudinov, “Isoperimetric inequalities for $l^p$-norms of the stress function of a multiply connected plane domain”, Russ. Math. Iz. VUZ, 57:9 (2013), 62–66 | DOI | MR | Zbl

[12] R.G. Salahudinov., “Isoperimetric inequalities for $l^p$-norms of the distance function to the boundary”, Uchen. Zap. Kazan. Univ. Ser. Fiz.-Mat. Nauki, 148, no. 2, 2006, 151–162 (in Russian) | Zbl

[13] J. Maly, D. Swanson, W. Ziemer, “The coarea formula for sobolev mappings”, Trans. Amer. Math. Soc., 355:01 (2002), 477–492 | MR

[14] R.G. Salakhudinov, “Payne type inequalities for $l^p$-norms of the warping functions”, J. Math. Anal. Appl., 410:2 (2014), 659–669 | DOI | MR | Zbl