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European Mathematical Society Publishing House
2016-09-19 17:05:47
Revista Matemática Iberoamericana
Rev. Mat. Iberoamericana
RMI
0213-2230
2235-0616
General
10.4171/RMI
http://www.ems-ph.org/doi/10.4171/RMI
subscribers
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society (from 2012)
29
2013
2
On isoperimetric inequalities with respect to infinite measures
Friedemann
Brock
Universität Leipzig, LEIPZIG, GERMANY
Anna
Mercaldo
Università degli Studi di Napoli “Federico II”, NAPOLI, ITALY
Maria Rosaria
Posteraro
Università degli Studi di Napoli “Federico II”, NAPOLI, ITALY
Isoperimetric inequalities, infinite measures, Steiner symmetrization, Schwarz symmetrization, comparison result, Pólya–Szegö inequality
We study isoperimetric problems with respect to infinite measures on $\mathbb{R} ^n$. In the case of the measure $\mu$ defined by $d\mu = e^{c|x|^2 }\, dx$, $c\geq 0$, we prove that, among all sets with given $\mu$-measure, the ball centered at the origin has the smallest (weighted) $\mu$-perimeter. Our results are then applied to obtain Pólya–Szegö-type inequalities, Sobolev embedding theorems, and a comparison result for elliptic boundary value problems.
Real functions
Partial differential equations
Functional analysis
General
665
690
10.4171/RMI/734
http://www.ems-ph.org/doi/10.4171/RMI/734