Rendiconti del Seminario Matematico della Università di Padova


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Volume 139, 2018, pp. 241–260
DOI: 10.4171/RSMUP/139-10

Published online: 2018-06-06

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

Guoqing He[1] and Peibiao Zhao[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

In this paper we study the isoperimetric problem in a class of $x$-spherically symmetric sets in the Grushin space $\mathbb{R}^{h+1}$ 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 $\big\{(x,y)\in \mathbb{R}^{h+1}\colon |y|<\int_{\mathrm {arcsin}|x|}^{\frac{\pi}{2}}\sin^{\alpha+1}(t)dt,\ |x| < 1\big\}$.

Keywords: Grushin space, density, weighted isoperimetric problem

He Guoqing, Zhao Peibiao: The isoperimetric problem in the Grushin space $\mathbb{R}^{h+1}$ with density $|x|^p$. Rend. Sem. Mat. Univ. Padova 139 (2018), 241-260. doi: 10.4171/RSMUP/139-10