Lower Bounds of Dirichlet Eigenvalues for General Grushin Type Bi-Subelliptic Operators (Analysis in Theory and Applications, 2022)
Let $Ω$ be a bounded open domain in ${\mathbb{R}}^{n}$ with smooth boundary $\partial Ω$. Let $X=(X_{1},X_{2},…,X_{m})$ be a system of general Grushin type vector fields defined on $Ω$ and the boundary $\partialΩ$ is non-characteristic for $X$. For $Δ _{X}=\sum_{j=1}^mX_j^2$, we denote $λ_{k}$ as the $k$-th eigenvalue for the bi-subelliptic operator $Δ _{X}^2$ on $Ω$. In this paper, by using the sharp sub-elliptic estimates and maximally hypoelliptic estimates, we give the optimal lower bound estimates of $λ_k$ for the operator $Δ _{X}^2$.