The first Grushin eigenvalue on cartesian product domains

In this paper, we consider the first eigenvalue λ₁(Ω) of the Grushin operator Δ_G := Δ_x1 + |x₁|^2sΔ_x2 with Dirichlet boundary conditions on a bounded domain Ω of ℝ^d = ℝ^d1+d2. We prove that λ₁(Ω) admits a unique minimizer in the class of domains with prescribed finite volume, which are the cartesian product of a set in ℝ^d1 and a set in ℝ^d2, and that the minimizer is the product of two balls Ω^∗₁ ⊆ ℝ^d1 and Ω^∗₂ ⊆ ℝ^d2. Moreover, we provide a lower bound for |Ω^∗₁ | and for λ₁(Ω^∗₁ × Ω^∗₂ ). Finally, we consider the limiting problem as s tends to 0 and to +∞.