TY - JOUR T1 - Lower Bounds of Dirichlet Eigenvalues for General Grushin Type Bi-Subelliptic Operators JO - Analysis in Theory and Applications VL - 1 SP - 66 EP - 84 PY - 2019 DA - 2019/04 SN - 35 DO - http://doi.org/10.4208/ata.OA-0002 UR - https://global-sci.org/intro/article_detail/ata/13092.html KW - Eigenvalues, degenerate elliptic operators, sub-elliptic estimate, maximally hypoelliptic estimate, bi-subelliptic operator. AB -

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 $\Delta _{X}^2$.