Some isoperimetric inequalities in the plane with radial power weights

TL;DR

This paper investigates isoperimetric inequalities in the plane with radial power weights, identifying conditions under which centered balls are uniquely optimal, extending classical results to weighted settings.

Contribution

It establishes new conditions on radial weights for which centered balls are the unique isoperimetric solutions, generalizing classical inequalities.

Findings

Centered balls are uniquely isoperimetric under specified weight conditions.

Conditions involve inequalities relating lpha, eta, and their combinations.

Classical case lpha=eta=0 is excluded from these results.

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 $\alpha-\beta+1>0$, $\alpha\leq 2\beta$ and $\alpha(\beta+1)\leq\beta^2$ except in the case $\alpha=\beta=0$ which corresponds to the classical isoperimetric inequality.