On weighted isoperimetric inequalities with non-radial densities

ABSTRACT

We consider a class of isoperimetric problems on ${\mathbb{R}}_{+}^N$, where the volume and the area element carry two different weights of the type $|x{|}^l{x}_N^{\alpha }$. We solve them in a special case while a more detailed study is contained in Alvino et al. (The isoperimetric problem for a class of non-radial weights and applications, in preparation). Our results imply a weighted Pólya–Szëgo principle and a priori estimates for weak solutions to a class of boundary value problems for degenerate elliptic equations.

KEYWORDS:

• Isoperimetric inequality
• weighted rearrangement
• norm inequality
• elliptic boundary value problem

2000 Mathematics Subject Classifications:

• 51M16
• 46E35
• 46E30
• 35P15

Acknowledgments

The authors are grateful to the referees for useful suggestions and comments. They also thank the University of Naples Federico II and South China University of Technology of Guangzhou for supporting visiting appointments and for their kind hospitality.

Disclosure statement

No potential conflict of interest was reported by the authors.