Lower bounds of Dirichlet eigenvalues for some degenerate elliptic operators
Lower bounds of Dirichlet eigenvalues for some degenerate elliptic operators
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DOI：
10.1007/s00526-015-0885-3
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发表时间：
2015-07
影响因子：
2.1
通讯作者：
Hua Chen;Peng Luo
中科院分区：
文献类型：
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作者：
Hua Chen;Peng Luo
设是一个具有光滑边界的有界开域，是一个定义在上的真实的光滑向量场系统，其边界对于X是非特征的。如果X满足Hörmander条件，则向量场是有限退化的，平方和算子是一个n阶退化椭圆算子，否则称之为无限退化算子.如果是的第j个Dirichlet特征值，则本文将研究的下界估计。首先，直接利用次椭圆估计，对一般的多项式退化的多项式递增的inj，给出了一个简单的下界估计。其次，如果是所谓的Grushin型退化椭圆算子，那么我们可以给出精确的下界估计。最后，利用对数正则性估计，证明了对于无限退化椭圆算子，的下界估计是对数递增的。
Letbe a bounded open domain inwith smooth boundary andbe a system of real smooth vector fields defined onwith the boundarywhich is non-characteristic forX. IfXsatisfies the Hörmander’s condition, then the vector fields is finite degenerate and the sum of square operatoris a finitely degenerate elliptic operator, otherwise the operatoris called infinitely degenerate. Ifis thejth Dirichlet eigenvalue foron, then this paper shall study the lower bound estimates for. Firstly, by using the sub-elliptic estimate directly, we shall give a simple lower bound estimates offor general finitely degeneratewhich is polynomial increasing inj. Secondly, ifis so-called Grushin type degenerate elliptic operator, then we can give a precise lower bound estimates for. Finally, by using logarithmic regularity estimate, for infinitely degenerate elliptic operatorwe prove that the lower bound estimates ofwill be logarithmic increasing inj.