Year: 2019
Analysis in Theory and Applications, Vol. 35 (2019), Iss. 1 : pp. 66–84
Abstract
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 $\Delta _{X}^2$.
Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/ata.OA-0002
Analysis in Theory and Applications, Vol. 35 (2019), Iss. 1 : pp. 66–84
Published online: 2019-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 19
Keywords: Eigenvalues degenerate elliptic operators sub-elliptic estimate maximally hypoelliptic estimate bi-subelliptic operator.
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