## Estimates of Dirichlet Eigenvalues for Degenerate Elliptic Operators

Title：Estimates of Dirichlet Eigenvalues for Degenerate Elliptic Operators

Abstract：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 $\partial\Omega$ 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 $\triangle_{X}=\sum_{j=1}^{m}X_j^2$ is a finitely degenerate elliptic operator, otherwise the operator $-\triangle_{X}$ is called infinitely  degenerate. If  $\lambda_j$ is the $j^{th}$ Dirichlet eigenvalue for $-\triangle_{X}$ 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 $\triangle_{X}$ which is polynomial increasing in $j$. Secondly, if $\triangle_{X}$ 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 $\triangle_{X}$ we prove that the lower bound estimates of $\lambda_j$ will be logarithmic increasing in $j$.