Lower bounds of Dirichlet eigenvalues for some degenerate elliptic operators

Abstract

Let $\Omega$ be a bounded open domain in $\mathbb {R}^n$ with smooth boundary and $X=(X_1, X_2, \ldots , X_m)$ be a system of real smooth vector fields defined on $\Omega$ with the boundary $\partial \Omega$ which is non-characteristic for X. If X satisfies the Hörmander’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 jth Dirichlet eigenvalue for $-\triangle _{X}$ on $\Omega$, then this paper 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.