We consider a class of isoperimetric problems on $R_{+}^{2}$ with respect to monomial weights. Let $\alpha ,\beta $ be real numbers such that $0\leq \alpha <\beta +1$, $\beta \leq 2\alpha$. We provide an explicit solution to the problem of minimizing the weighted perimeter $\int_{\partial \Omega }y^{\beta }dxdy$, among all smooth sets $\Omega $ in $R_{+}^{2}$ with fixed weighted measure $\int\limits_{\Omega }y^{\beta }dxdy$. Our results also imply an estimate of a weighted Cheeger constant and a bound for a class of nonlinear elliptic PDE's.
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