JournalsrmiVol. 29, No. 2pp. 665–690

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
On isoperimetric inequalities    with respect to infinite measures  cover

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