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 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 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.