The isoperimetric problem in the Grushin space $\mathbb{R}^{h+1}$ with density $|x|^p$

He, Guoqing; Zhao, Peibiao

doi:10.4171/RSMUP/139-10

The isoperimetric problem in the Grushin space

ℝ^{h + 1}

with density

{| x |}^{p}

He, Guoqing ¹ ; Zhao, Peibiao ²

¹ Anhui Normal University, Wuhu City, China and Nanjing University of Science and Technology, Nanjing, China
² Nanjing University of Science and Technology, Nanjing, China

Rendiconti del Seminario Matematico della Università di Padova, Tome 139 (2018), pp. 241-260.

Résumé

In this paper we study the isoperimetric problem in a class of $x$ -spherically symmetric sets in the Grushin space $ℝ^{h + 1}$ with density ${| x |}^{p}$ , $p > - h + 1$ . First we prove the existence of weighted isoperimetric sets. Then we deduce that, up to a vertical translation, a dilation and a negligible set, the weighted isoperimetric set is only of the form $\{(x, y) \in ℝ^{h + 1} : | y | < \int_{\arcsin | x |}^{\frac{π}{2}} {sin}^{α + 1} (t) d t, | x | < 1\}$ .

MR Zbl

DOI : 10.4171/RSMUP/139-10

Classification : 53, 49
Mots clés : Grushin space, density, weighted isoperimetric problem

Affiliations des auteurs :

He, Guoqing ¹ ; Zhao, Peibiao ²

¹ Anhui Normal University, Wuhu City, China and Nanjing University of Science and Technology, Nanjing, China
² Nanjing University of Science and Technology, Nanjing, China

@article{RSMUP_2018__139__241_0,
     author = {He, Guoqing and Zhao, Peibiao},
     title = {The isoperimetric problem in the {Grushin} space $\mathbb{R}^{h+1}$ with density $|x|^p$},
     journal = {Rendiconti del Seminario Matematico della Universit\`a di Padova},
     pages = {241--260},
     publisher = {European Mathematical Society Publishing House},
     address = {Zuerich, Switzerland},
     volume = {139},
     year = {2018},
     doi = {10.4171/RSMUP/139-10},
     mrnumber = {3825189},
     zbl = {1406.53037},
     url = {http://archive.numdam.org/articles/10.4171/RSMUP/139-10/}
}

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AU  - Zhao, Peibiao
TI  - The isoperimetric problem in the Grushin space $\mathbb{R}^{h+1}$ with density $|x|^p$
JO  - Rendiconti del Seminario Matematico della Università di Padova
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%A Zhao, Peibiao
%T The isoperimetric problem in the Grushin space $\mathbb{R}^{h+1}$ with density $|x|^p$
%J Rendiconti del Seminario Matematico della Università di Padova
%D 2018
%P 241-260
%V 139
%I European Mathematical Society Publishing House
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%U http://archive.numdam.org/articles/10.4171/RSMUP/139-10/
%R 10.4171/RSMUP/139-10
%F RSMUP_2018__139__241_0

He, Guoqing; Zhao, Peibiao. The isoperimetric problem in the Grushin space $\mathbb{R}^{h+1}$ with density $|x|^p$. Rendiconti del Seminario Matematico della Università di Padova, Tome 139 (2018), pp. 241-260. doi : 10.4171/RSMUP/139-10. http://archive.numdam.org/articles/10.4171/RSMUP/139-10/

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