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

Abstract

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