The first Grushin eigenvalue on cartesian product domains

Abstract In this paper, we consider the first eigenvalue λ 1 ⁢ ( Ω stretchy="false">) {\lambda_{1}(\Omega)} of Grushin operator mathvariant="normal">Δ G := x + fence="true" stretchy="false">| 2 s {\Delta_{G}:=\Delta_{x_{1}}+\lvert x_{1}\rvert^{2s}\Delta_{x_{2}}} with Dirichlet boundary conditions on a bounded domain Ω ℝ d = {\mathbb{R}^{d}=\mathbb{R}^{d_{1}+d_{2}}} . We prove that admits unique minimizer in class domains prescribed finite volume, which are cartesian product set {\mathbb{R}^{d_{1}}} and {\mathbb{R}^{d_{2}}} , is two balls * ⊆ {\Omega^{*}_{1}\subseteq\mathbb{R}^{d_{1}}} {\Omega_{2}^{*}\subseteq\mathbb{R}^{d_{2}}} Moreover, provide lower bound for {\lvert\Omega^{*}_{1}\rvert} × {\lambda_{1}(\Omega_{1}^{*}\times\Omega_{2}^{*})} Finally, limiting problem as s tends to 0 mathvariant="normal">∞ {+\infty}