The ε-ε property and the boundedness of isoperimetric sets with different monomial weights

¹ ICEx - Departamento de Matemática, Universidade Federal de Minas Gerais, Av. Antônio Carlos 6627, Pampulha, CP 702, Belo Horizonte 30161-970, MG, Brazil
² Departamento de Matemática, Universidade Federal de Roraima, Av. Cap. Ene Garcêz 2413, Aeroporto, Boa Vista 69310-000, RR, Brazil
³ Centro de Ciências Exatas e Tecnológicas, Universidade Federal do Acre, Campus Universitário, BR 364, Km 04, Distrito industrial, Rio Branco 39012-510 AC, Brazil

^* Corresponding author: eabreu@ufmg.br

Received: 1 November 2022
Accepted: 26 March 2023

Abstract

We consider a class of monomial weights 𝑥^A = |𝑥₁|^𝑎₁…|𝑥_N|^𝑎N in ℝ^N, where a_i is a nonnegative real number for each i ∈ {1,…,N}, and we establish the ε — ε property and the boundedness of isoperimetric sets with different monomial weights for the perimeter and volume. Moreover, we present cases of nonexistence of the isoperimetric inequality when it is not possible to associate the corresponding Sobolev inequality. Finally, for N = 2, we developed an original type of symmetrization, which we call star-shaped Steiner symmetrization, and we apply it to a class of isoperimetric problems with different monomial weights.