# Some weighted isoperimetric problems on $\mathbb{R}^N _+ $ with stable half balls have no solutions

## TL;DR

This paper demonstrates that certain weighted isoperimetric problems in half-spaces, even with stable half-balls, can lack solutions, highlighting counter-intuitive phenomena in geometric measure theory.

## Contribution

It reveals that weighted isoperimetric problems with stable half-balls can have no solutions, providing new insights into stability and nonexistence in weighted geometric problems.

## Key findings

- Half-balls centered at the origin are stable for certain weights.
- Some weighted isoperimetric problems have no solutions despite stability.
- Results extend to weighted problems in the entire space.

## Abstract

We show the counter-intuitive fact that some weighted isoperimetric problems on the half-space $ \mathbb{R}^N _+ $, for which half-balls centered at the origin are stable, have no solutions. A particular case is the measure $d\mu = x_N ^{\alpha } \, dx$, with $\alpha \in (-1,0)$. Some results on stability and nonexistence for weighted isoperimetric problems on $\mathbb{R}^N $ are also obtained.