Estimates of Dirichlet Eigenvalues for Degenerate Elliptic Operators (Workshop on the Boltzmann Equation, Microlocal Analysis and Related Topics)

Abstract

Let Ω be a bounded open domain in Rn with smooth boundary and X = (X1, X2, …, Xm) be a system of real smooth vector fields defined on Ω with the boundary ∂Ω which is non-characteristic for X. If X satisfies the H ormander's condition, then the vector fields is finite degenerate and the sum of square operator △X = Σm j=1 X2 j is a finitely degenerate elliptic operator, otherwise the operator -△X is called infinitely degenerate. If λj is the jth Dirichlet eigenvalue for -△X on Ω, then this paper shall study the lower bound estimates for λj. Firstly, by using the sub-elliptic estimate directly, we shall give a simple lower bound estimates of λj for general finitely degenerate △X which is polynomial increasing in j. Secondly, if △X is so-called Grushin type degenerate elliptic operator, then we can give a precise lower bound estimates for λj. Finally, by using logarithmic regularity estimate, for infinitely degenerate elliptic operator △X we prove that the lower bound estimates of λj will be logarithmic increasing in j.