Isoperimetric problems in sectors with density

TL;DR

This paper investigates the isoperimetric problem in sectors with density functions and characterizes solutions based on sector angles, also solving related problems in higher-dimensional spaces with specific densities.

Contribution

It provides a comprehensive characterization of isoperimetric solutions in sectors with density and extends the analysis to higher dimensions with negative density powers.

Findings

Solutions depend on sector angle and density parameters.

Explicit solutions are characterized for specific density functions.

The problem is extended to higher dimensions with negative density powers.

Abstract

We consider the isoperimetric problem in planar sectors with density $r^{p}$, and with density $a>1$ inside the unit disk and $1$ outside. We characterize solutions as a function of sector angle. We also solve the isoperimetric problem in $\mathbb{R}^{n}$ with density $r^{p},\; p<0$.