The first Grushin eigenvalue on cartesian product domains

Abstract

In this paper we consider the first eigenvalue λ 1 (Ω) of the Grushin operator ∆ G := ∆x 1 + |x 1 | 2s ∆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 +∞.