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The first Grushin eigenvalue on cartesian product domains.

The first Grushin eigenvalue on cartesian product domains.

Authors :
Luzzini, Paolo
Provenzano, Luigi
Stubbe, Joachim
Source :
Advances in Calculus of Variations. Apr2024, Vol. 17 Issue 2, p353-371. 19p.
Publication Year :
2024

Abstract

In this paper, we consider the first eigenvalue λ 1 ⁢ (Ω) of the Grushin operator Δ G := Δ x 1 + | x 1 | 2 ⁢ s ⁢ Δ x 2 with Dirichlet boundary conditions on a bounded domain Ω of ℝ d = ℝ d 1 + d 2 . We prove that λ 1 ⁢ (Ω) admits a unique minimizer in the class of domains with prescribed finite volume, which are the cartesian product of a set in ℝ d 1 and a set in ℝ d 2 , and that the minimizer is the product of two balls Ω 1 * ⊆ ℝ d 1 and Ω 2 * ⊆ ℝ d 2 . Moreover, we provide a lower bound for | Ω 1 * | and for λ 1 ⁢ (Ω 1 * × Ω 2 *) . Finally, we consider the limiting problem as s tends to 0 and to + ∞ . [ABSTRACT FROM AUTHOR]

Subjects

Subjects :
*EIGENVALUES
*SCHRODINGER operator

Details

Language :
English
ISSN :
18648258
Volume :
17
Issue :
2
Database :
Academic Search Index
Journal :
Advances in Calculus of Variations
Publication Type :
Academic Journal
Accession number :
176386411
Full Text :
https://doi.org/10.1515/acv-2022-0015