Inequalities of Dirichlet eigenvalues for degenerate elliptic partial differential operators

Abstract

Let X j , Y j (j = 1, • • •, n) be vector fields satisfying Hörmander's condition andIn this paper, we establish some inequalities of Dirichlet eigenvalues for degenerate elliptic partial differential operator ∆ L and ∆ 2 L . These inequalities extend Yang's inequalities for Dirichlet eigenvalues of Laplacian to the settings here and the forms of inequalities are more general than Yang's inequalities. To obtain them, we give a generalization of the inequality by Chebyshev.