Payne-polya-weinberger Type Inequalities for Eigenvalues of Nonelliptic Operators

Let denote the Laplacian in the Euclidean space. The classic upper estimates, independent of the domain, for the gaps of eigenvalues of − , (− )2 and (− )k(k ≥ 3) were studied extensively by many mathematicians, cf. Payne, Polya and Weinberger [16], Hile and Yeh [10], Chen and Qian [2], Guo [8] etc.. The asymptotic behaviors of eigenvalues for degenerate elliptic operators were considered by Beals,Greiner and Stanton [1], Menikoff and Sjöstrand [15], Fefferman and Phone [3, 4], Garofalo and Shen [7], respectively. In this paper, we are concerned with the following eigenvalue problem − Hn u = λu, in Ω, u = 0, on ∂Ω (1.1)