# The isoperimetric problem in the Heisenberg group Hn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pmb {\mathbb {H}^n}$$\end{document} with density

The isoperimetric problem in the Heisenberg group Hn\documentclass[12pt]{minimal}... We study the isoperimetric problem for the axially symmetric sets in the Heisenberg group Hn\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb {H}^n$$\end{document} with density |z|p\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$|z|^p$$\end{document}. At first, we prove the existence of weighted isoperimetric sets. Then, we characterize weighted isoperimetric sets uniquely as bubble sets. Finally, we deduce an interesting result that, up to a constant multiplicator, |z|p\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$|z|^p$$\end{document} is the only horizontal radial density for which bubble sets can be weighted isoperimetric sets. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Analysis and Mathematical Physics Springer Journals

# The isoperimetric problem in the Heisenberg group Hn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pmb {\mathbb {H}^n}$$\end{document} with density

, Volume 10 (2) – May 17, 2020
23 pages

/lp/springer-journals/the-isoperimetric-problem-in-the-heisenberg-group-hn-documentclass-i30rL2HO1m
Publisher
Springer Journals
Copyright © Springer Nature Switzerland AG 2020
ISSN
1664-2368
eISSN
1664-235X
DOI
10.1007/s13324-020-00367-2
Publisher site
See Article on Publisher Site

### Abstract

We study the isoperimetric problem for the axially symmetric sets in the Heisenberg group Hn\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb {H}^n$$\end{document} with density |z|p\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$|z|^p$$\end{document}. At first, we prove the existence of weighted isoperimetric sets. Then, we characterize weighted isoperimetric sets uniquely as bubble sets. Finally, we deduce an interesting result that, up to a constant multiplicator, |z|p\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$|z|^p$$\end{document} is the only horizontal radial density for which bubble sets can be weighted isoperimetric sets.

### Journal

Analysis and Mathematical PhysicsSpringer Journals

Published: May 17, 2020