The first Grushin eigenvalue on cartesian product domains

In this paper we consider the first eigenvalue $\lambda_1(\Omega)$ of the Grushin operator $\Delta_G:=\Delta_{x_1}+|x_1|^{2s}\Delta_{x_2}$ with Dirichlet boundary conditions on a bounded domain $\Omega$ of $\mathbb{R}^d= \mathbb{R}^{d_1+d_2}$. We prove that $\lambda_1(\Omega)$ 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 $\Omega^*_1 \subseteq \mathbb{R}^{d_1}$ and $\Omega_2^* \subseteq \mathbb{R}^{d_2}$. Moreover, we provide a lower bound for $|\Omega^*_1|$ and for $\lambda_1(\Omega_1^*\times\Omega_2^*)$. Finally, we consider the limiting problem as $s$ tends to $0$ and to $+\infty$.