The first Grushin eigenvalue on cartesian product domains

In this paper we consider the first eigenvalue $λ_1(Ω)$ of the Grushin operator $Δ_G:=Δ_{x_1}+|x_1|^{2s}Δ_{x_2}$ with Dirichlet boundary conditions on a bounded domain $Ω$ of $\mathbb{R}^d= \mathbb{R}^{d_1+d_2}$. 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 $\mathbb{R}^{d_1}$ and a set in $\mathbb{R}^{d_2}$, and that the minimizer is the product of two balls $Ω^*_1 \subseteq \mathbb{R}^{d_1}$ and $Ω_2^* \subseteq \mathbb{R}^{d_2}$. Moreover, we provide a lower bound for $|Ω^*_1|$ and for $λ_1(Ω_1^*\timesΩ_2^*)$. Finally, we consider the limiting problem as $s$ tends to $0$ and to $+\infty$.

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