Volume 27, 2021Regular articles published in advance of the transition of the journal to Subscribe to Open (S2O). Free supplement sponsored by the Fonds National pour la Science Ouverte
|Number of page(s)||29|
|Published online||01 March 2021|
Some isoperimetric inequalities with respect to monomial weights
Università di Napoli Federico II, Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Complesso Monte S. Angelo, via Cintia,
2 University of Rostock, Institute of Mathematics, 18057 Rostock, Ulmenstr. 69, Germany.
* Corresponding author: firstname.lastname@example.org
Accepted: 28 July 2020
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 ∬Ωyβdxdy, the weighted perimeter ∫∂Ωyα 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.
Mathematics Subject Classification: 51M16 / 46E35 / 46E30 / 35P15
Key words: Isoperimetric inequality / weighted Cheeger set / eigenvalue problems
© EDP Sciences, SMAI 2021
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