Lower bounds of eigenvalues for a class of bi-subelliptic operators
Lower bounds of eigenvalues for a class of bi-subelliptic operators
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一类双亚椭圆算子的特征值下界
DOI：
10.1016/j.jde.2017.02.018
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发表时间：
2017
影响因子：
2.4
通讯作者：
Zhou Yifu
中科院分区：
文献类型：
--
作者：
Chen Hua;Zhou Yifu
设Ω是Rn中一个有界开域，其边界是光滑的，X=（X1，X2，X2，Xm）是定义在Ω上的一个真实的光滑向量场系统，其边界是X的非特征的.若X满足Hörmander条件，则向量场是n阶退化的，且平方算子和△ X=∑ i= 1mXi2是次椭圆算子.设λ k是Ω上双次椭圆算子△ X2的第k个特征值.本文引入广义Métivier条件，研究了满足Hörmander条件或广义Métivier条件的一类退化向量场系统X上算子△ X2的Dirichlet特征值的下界.利用次椭圆估计，我们给出了λ k的一个显式下界估计，它是关于Hörmander指数或广义Métivier指数的阶的关于k的多项式递增的.
Let Ω be a bounded open domain in R n with smooth boundary and X=(X 1, X 2,⋯, X m) be a system of real smooth vector fields defined on Ω with the boundary∂ Ω which is non-characteristic for X. If X satisfies the Hörmander's condition, then the vector fields are finitely degenerate and the sum of square operators△ X=∑ i= 1 m X i 2 is a subelliptic operator. Let λ k be the k-th eigenvalue for the bi-subelliptic operator△ X 2 on Ω. In this paper, we introduce the generalized Métivier's condition and study the lower bounds of Dirichlet eigenvalues for the operator△ X 2 on some finitely degenerate systems of vector fields X which satisfy the Hörmander's condition or the generalized Métivier's condition. By using the subelliptic estimates, we shall give a explicit lower bound estimates of λ k which is polynomial increasing in k with the order relating to the Hörmander index or the generalized Métivier index.