Isoperimetric Inequality for degenerate elliptic operators of Grushin type

TL;DR

This paper establishes an isoperimetric inequality for a class of degenerate elliptic operators of Grushin type, linking volume and perimeter in the associated geometric space.

Contribution

It proves a new isoperimetric inequality for Grushin-type operators, extending geometric analysis to degenerate elliptic operators with explicit constants.

Findings

Derived the isoperimetric inequality for Grushin spaces.

Identified the critical exponent Q relating volume and perimeter.

Established bounds for smooth bounded domains in the Grushin setting.

Abstract

Let $n,m\ge 1$, $\alpha\in(0,1)$, and $\beta\ge 0$. For the Grushin-type operator \[ L=-\nabla_x\!\cdot\!\bigl(|x|^{2\alpha}\nabla_x\bigr)+|x|^{2\beta}\Delta_y \qquad \text{on } \mathbb R^n\times \mathbb R^m, \] we prove the isoperimetric inequality on the associated Grushin space. Equivalently, if \[ Q=\frac{n+m(\beta+1-\alpha)}{1-\alpha}, \] then \[ |\Omega|^{\frac{Q-1}{Q}}\le C\,P(\Omega) \] for every smooth bounded domain $\Omega\subset \mathbb R^{n+m}$.