Anal. Theory Appl., 35 (2019), pp. 66-84.
Published online: 2019-04
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Let $\Omega$ be a bounded open domain in ${\mathbb{R}}^{n}$ with smooth boundary $\partial \Omega$. Let $X=(X_{1},X_{2},\cdots,X_{m})$ be a system of general Grushin type vector fields defined on $\Omega$ and the boundary $\partial\Omega$ is non-characteristic for $X$. For $\Delta _{X}=\sum_{j=1}^mX_j^2$, we denote $\lambda_{k}$ as the $k$-th eigenvalue for the bi-subelliptic operator $\Delta _{X}^2$ on $\Omega$. In this paper, by using the sharp sub-elliptic estimates and maximally hypoelliptic estimates, we give the optimal lower bound estimates of $\lambda_k$ for the operator $\Delta _{X}^2$.
}, issn = {1573-8175}, doi = {https://doi.org/10.4208/ata.OA-0002}, url = {http://global-sci.org/intro/article_detail/ata/13092.html} }Let $\Omega$ be a bounded open domain in ${\mathbb{R}}^{n}$ with smooth boundary $\partial \Omega$. Let $X=(X_{1},X_{2},\cdots,X_{m})$ be a system of general Grushin type vector fields defined on $\Omega$ and the boundary $\partial\Omega$ is non-characteristic for $X$. For $\Delta _{X}=\sum_{j=1}^mX_j^2$, we denote $\lambda_{k}$ as the $k$-th eigenvalue for the bi-subelliptic operator $\Delta _{X}^2$ on $\Omega$. In this paper, by using the sharp sub-elliptic estimates and maximally hypoelliptic estimates, we give the optimal lower bound estimates of $\lambda_k$ for the operator $\Delta _{X}^2$.