# Some isoperimetric inequalities with respect to monomial weights

## TL;DR

This paper establishes isoperimetric inequalities for weighted measures in the upper half-plane, identifying explicit minimizers and deriving implications for weighted Cheeger constants and eigenvalue bounds.

## Contribution

It provides explicit solutions to a class of weighted isoperimetric problems with monomial weights, extending classical results to weighted settings.

## Key findings

- Weighted perimeter minimized by symmetric sets
- Explicit form of optimal sets derived
- Implications for Cheeger constants and eigenvalues

## Abstract

We solve a class of isoperimetric problems on $\mathbb{R}^2_+ :=\left\{ (x,y)\in \mathbb{R} ^2 : y>0 \right\}$ with respect to monomial weights. Let $\alpha $ and $\beta $ be real numbers such that $0\le \alpha <\beta+1$, $\beta\le 2 \alpha$. We show that, among all smooth sets $\Omega$ in $\mathbb{R} ^2_+$ with fixed weighted measure $\iint_{\Omega } y^{\beta} dxdy$, the weighted perimeter $\int_{\partial \Omega } y^\alpha \, ds$ achieves its minimum for a smooth set which is symmetric w.r.t. to the $y$--axis, and is explicitly given. Our results also imply an estimate of a weighted Cheeger constant and a lower bound for the first eigenvalue of a class of nonlinear problems.