Journal of Inequalities and Applications 
Volume 4 (1999), Issue 3, Pages 215-240
doi:10.1155/S1025583499000375

A weighted isoperimetric inequality and applications to symmetrization

M. F. Betta,1 F. Brock,2 A. Mercaldo,3 and M. R. Posteraro3

 1Dipartimento di Matematica, Seconde Universita di Napoli, Piazza Duamo, Caserte 81100, Italy
2Fakultät für Mathematik und Informatik, Universität Leipzig, Augustusplatz 10, Leipzig D 04109, Germany
3Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Universita di Napoli, Complesso Monte S. Angelo, Via Cintia, Napoli 80126, Italy
Received 20 October 1998; Revised 15 January 1999

Abstract

We prove an inequality of the form ∫∂Ωa(|x|)ℋn−1(dx)≥∫∂Ba(|x|)ℋn−1(dx), where Ω is a bounded domain in Rn with smooth boundary, B is a ball centered in the origin having the same measure as Ω. From this we derive inequalities comparing a weighted Sobolev norm of a given function with the norm of its symmetric decreasing rearrangement. Furthermore, we use the inequality to obtain comparison results for elliptic boundary value problems.