The first Grushin eigenvalue on cartesian product domains

Abstract

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