Inequalities of Dirichlet eigenvalues for degenerate elliptic partial differential operators

Abstract

Let $X_j,Y_j(j=1,\cdot \cdot \cdot ,n)$ be vector fields satisfying Hörmander's condition and ${\Delta }_L=\sum _{j=1}^n(X_j^2+Y_j^2)$. In this paper, we establish some inequalities of Dirichlet eigenvalues for degenerate elliptic partial differential operator ${\Delta }_L$ and ${\Delta }_L^2$. 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.