Lower Bounds of Dirichlet Eigenvalues for General Grushin Type Bi-Subelliptic Operators

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.