Some isoperimetric inequalities with respect to monomial weights

Abstract

We solve a class of isoperimetric problems on ${\mathbb{R}}_{+}^2:=\left\{(x,y)\in {\mathbb{R}}^2:y>0\right\}$ with respect to monomial weights. Let $\alpha$ and $\beta$ be real numbers such that $0\le \alpha <\beta +1$, $\beta \le 2\alpha$. We show that, among all smooth sets $\Omega$ in ${\mathbb{R}}_{+}^2$ with fixed weighted measure ${\iint }_{\Omega }{y}^{\beta }dxdy$, the weighted perimeter ${\int }_{\partial \Omega }{y}^{\alpha }ds$ achieves its minimum for a smooth set which is symmetric w.r.t. to the $y$--axis, and is explicitly given. Our results also imply an estimate of a weighted Cheeger constant and a lower bound for the first eigenvalue of a class of nonlinear problems.