We study isoperimetric problems with respect to infinite measures on . In the case of the measure defined by , , we prove that, among all sets with given -measure, the ball centered at the origin has the smallest (weighted) -perimeter. Our results are then applied to obtain Pólya–Szegö-type inequalities, Sobolev embedding theorems, and a comparison result for elliptic boundary value problems.
Cite this article
Friedemann Brock, Anna Mercaldo, Maria Rosaria Posteraro, On isoperimetric inequalities with respect to infinite measures . Rev. Mat. Iberoam. 29 (2013), no. 2, pp. 665–690DOI 10.4171/RMI/734