Let $Omega$ be a bounded open domain in $R^n$ with smooth boundary and $X=(X_1, X_2, cdots, X_m)$ be a system of real smooth vector fields defined on $Omega$ with smooth boundary $partialOmega$ which is non-characteristic for $X$. If $X$ satisfies the Hormander’s condition, then the vector fields is finite degenerate and the sum of square operator $ riangle_{X}=sum_{j=1}^{m}X_j^2$ is a finitely degenerate elliptic operator, otherwise the operator $- riangle_{X}$ is called infinitely degenerate. If $lambda_j$ is the $j^{th}$ Dirichlet eigenvalue for $- riangle_{X}$ on $Omega$, then in this talk, we shall study the lower bound estimates for $lambda_j$. Firstly, by using the sub-elliptic estimate directly, we shall give a simple lower bound estimates of $lambda_j$ for general finitely degenerate $ riangle_{X}$ which is polynomial increasing in $j$. Secondly, if $ riangle_{X}$ is so-called Grushin type degenerate elliptic operator, then we can give a precise lower bound estimates for $lambda_j$. Finally, by using logarithmic regularity estimate, for infinitely degenerate elliptic operator $ riangle_{X}$ we prove that the lower bound estimates of $lambda_j$ will be logarithmic increasing in $j$.
Prof. Chen completed his Ph.D. from WuHan University in 1986. Then he completed his Postdoctoral from University of Dundee. He is currently Head of School of Mathematics and Statistics at WuHan University. His main research interests include the microlocal analysis of partial differential equations, singular and degenerate partial differential equations. He has published in top journals such as For more information, please visit: Edited by: Shi Yajie |