Some isoperimetric inequalities in the plane with radial power weights

Abstract

We consider the punctured plane with volume density $|x{|}^{\alpha }$ and perimeter density $|x{|}^{\beta }$. We show that centred balls are uniquely isoperimetric for indices $(\alpha ,\beta )$ which satisfy the conditions $\alpha -\beta +1>0$, $\alpha \le 2\beta$ and $\alpha (\beta +1)\le {\beta }^2$ except in the case $\alpha =\beta =0$ which corresponds to the classical isoperimetric inequality.

arXiv comment56 pages, 9 figures