Some isoperimetric inequalities with respect to monomial weights

¹ Università di Napoli Federico II, Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Complesso Monte S. Angelo, via Cintia, 80126 Napoli, Italy.
² University of Rostock, Institute of Mathematics, 18057 Rostock, Ulmenstr. 69, Germany.

^* Corresponding author: mercaldo@unina.it

Received: 9 September 2020
Accepted: 28 July 2020

Abstract

We solve a class of isoperimetric problems on ℝ₊² with respect to monomial weights. Let α and β be real numbers such that 0 ≤ α < β + 1, β ≤ 2α. We show that, among all smooth sets Ω in ℝ₊² with fixed weighted measure ∬_Ωy^βdxdy, the weighted perimeter ∫_∂Ωy^α 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 bound for eigenvalues of some nonlinear problems.