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

He, Guoqing; Zhao, Peibiao

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

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 ${\left|x\right|}^p$, $p>-h+1$. 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 $\left\{(x,y)\in {ℝ}^{h+1}:\left|y\right|<{\int }_{\arcsin \left|x\right|}^{\frac{\pi }{2}}{sin}^{\alpha +1}\left(t\right)dt,\left|x\right|<1\right\}$.