Inequalities of Dirichlet eigenvalues for degenerate elliptic partial differential operators

TL;DR

This paper establishes generalized inequalities for Dirichlet eigenvalues of degenerate elliptic operators, extending Yang's inequalities from Laplacians to more complex operators under Hörmander's condition.

Contribution

It introduces new eigenvalue inequalities for degenerate elliptic operators, generalizing and extending Yang's inequalities to broader settings.

Findings

Inequalities for eigenvalues of ${ riangle_L}$ and ${ riangle_L}^2$ are established.

Results extend Yang's inequalities to degenerate elliptic operators.

Generalized Chebyshev inequality is used in derivations.

Abstract

Let ${X_j},{Y_j}(j = 1, \cdot \cdot \cdot,n)$ be vector fields satisfying H\"{o}rmander's condition and ${\Delta_L} = \sum\limits_{j = 1}^n {(X_j^2 + Y_j^2)}$. 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.