Isoperimetric inequalities for $l^p$-norms of the distance function to the boundary

R. G. Salahudinov

Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Kazanskii Gosudarstvennyi Universitet. Uchenye Zapiski. Seriya Fiziko-Matematichaskie Nauki, Tome 148 (2006) no. 2, pp. 151-162

Résumé

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.