论文标题
笛卡尔产品领域的第一个Grushin特征值
The first Grushin eigenvalue on cartesian product domains
论文作者
论文摘要
在本文中，我们考虑了Grushin操作员的第一个特征$λ_1（ω）$ $δ_g：=δ__{x_1}+| x_1 | x_1 |^{2S}δ_{x_2} $ cant dirichlet to dirichlet to dirichlet co. \ mathbb {r}^{d_1+d_2} $。我们证明，$λ_1（ω）$承认有限量的域类别中的独特最小化器，这是$ \ mathbb {r}^{d_1} $中的笛卡尔产物，在$ \ mathbb {rmathbb {r}^r}^{r}^d_2}^{d _2} $ sep y tw of tw of tw of balls usimizizizizizizer中， \ mathbb {r}^{d_1} $和$ω_2^* \ subseteq \ mathbb {r}^{d_2} $。此外，我们为$ |ω^*_ 1 | $和$λ_1（ω_1^*\timesΩ_2^*）$提供了下限。最后，我们将限制问题视为$ s $倾向于$ 0 $，而$+\ \ iffty $。
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 $+\infty$.
