On isoperimetric inequalities with respect to infinite measures

  • Friedemann Brock

    Universität Leipzig, Germany
  • Anna Mercaldo

    Università degli Studi di Napoli “Federico II”, Italy
  • Maria Rosaria Posteraro

    Università degli Studi di Napoli “Federico II”, Italy

Abstract

We study isoperimetric problems with respect to infinite measures on Rn\mathbb{R} ^n. In the case of the measure μ\mu defined by dμ=ecx2dxd\mu = e^{c|x|^2 }\, dx, c0c\geq 0, we prove that, among all sets with given μ\mu-measure, the ball centered at the origin has the smallest (weighted) μ\mu-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. 665–690

DOI 10.4171/RMI/734