| 摘要: |
| 本文考虑了如下具有Navier边界的紧致光滑度量测度空间上重漂移Laplace算子的buckling问题和夹板振动问题：
$$\left\{\begin{aligned}
&\Delta_f^2 u=-\Gamma\Delta_f u,&& \mbox{in}\;\;M,\\
&u=(1-\sigma)\dfrac{\partial^2 u}{\partial v^2}+\sigma \Delta u=0,&& \mbox{on}\;\;\partial M
\end{aligned}\right.$$
和
$$\left\{\begin{aligned}
&\Delta^2_f u-\tau\Delta_f u=\Gamma u,&& \mbox{in}\;\; M,\\
&u=(1-\sigma)\dfrac{\partial^2 u}{\partial v^2}+\sigma\Delta u=0,&& \mbox{on}\;\;\partial M,
\end{aligned}\right.$$
其中，$\tau$是一个非负常数，且$0\leq \sigma<1$. 我们获得了这两类问题的第一特征值的下界估计. 我们的结果包含了前人关于具有Dirichlet边界条件的重漂移Laplace算子的夹持薄板问题和buckling问题的相关结果. |
| 关键词: 重漂移Laplace算子； 第一特征值 Navier边界； buckling问题； 夹板振动问题 |
| DOI： |
| 分类号:O186.1 |
| 基金项目: |
|
| THE FIRST EIGENVALUES OF THE BI-DRIFTING LAPLACIAN WITH THE NAVIER BOUNDARY CONDITION |
|
Sun He-jun
|
|
Nanjing University of Science and Technology
|
| Abstract: |
| In this paper, we investigate the buckling problem and the vibration problem of the bi-drifting Laplacian $\Delta^2_f$ on a compact smooth metric measure space $(M,g,f)$ with the Navier boundary as follows
$$\left\{\begin{aligned}
&\Delta_f^2 u=-\Gamma\Delta_f u,&& \mbox{in}\;\;M,\\
&u=(1-\sigma)\dfrac{\partial^2 u}{\partial v^2}+\sigma \Delta u=0,&& \mbox{on}\;\;\partial M
\end{aligned}\right.$$
and
$$\left\{\begin{aligned}
&\Delta^2_f u-\tau\Delta_f u=\Gamma u,&& \mbox{in}\;\; M,\\
&u=(1-\sigma)\dfrac{\partial^2 u}{\partial v^2}+\sigma\Delta u=0,&&\mbox{on}\;\;\partial M,
\end{aligned}\right.$$
where $\tau$ is a non-negative constant and $0\leq \sigma<1$. We obtain some lower bounds for the first eigenvalues of these two problems. Moreover, our results cover some previous results for the clamped plate problem and buckling problem of the bi-drifting Laplacian with the Dirichlet boundary condition. |
| Key words: bi-drifting Laplacian the first eigenvalue Navier boundary buckling problem vibration problem |
