Volume 35, Issue 1
Lower Bounds of Dirichlet Eigenvalues for General Grushin Type Bi-Subelliptic Operators

Anal. Theory Appl., 35 (2019), pp. 66-84.

Published online: 2019-04

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• 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 $\Delta _{X}^2$.

• Keywords

Eigenvalues, degenerate elliptic operators, sub-elliptic estimate, maximally hypoelliptic estimate, bi-subelliptic operator.

35J30, 35J70, 35P15

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@Article{ATA-35-66, author = {}, title = {Lower Bounds of Dirichlet Eigenvalues for General Grushin Type Bi-Subelliptic Operators}, journal = {Analysis in Theory and Applications}, year = {2019}, volume = {35}, number = {1}, pages = {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 $\Delta _{X}^2$.

}, issn = {1573-8175}, doi = {https://doi.org/10.4208/ata.OA-0002}, url = {http://global-sci.org/intro/article_detail/ata/13092.html} }
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 $\Delta _{X}^2$.

Hua Chen, Hongge Chen, Junfang Wang & Nana Zhang. (2020). Lower Bounds of Dirichlet Eigenvalues for General Grushin Type Bi-Subelliptic Operators. Analysis in Theory and Applications. 35 (1). 66-84. doi:10.4208/ata.OA-0002
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