完备黎曼流形上四阶散度型椭圆算子的特征值估计

Estimates for Eigenvalues of Fourth-order Elliptic Operators in Divergence Form on Complete Riemanian Manifolds

研究了完备黎曼流形上四阶散度型椭圆算子的特征值问题,得到了特征值的一个基本不等式.由这个基本不等式,得到具有特殊函数的完备黎曼流形上四阶散度型椭圆算子的特征值估计的万有不等式,同时给出具有这些特殊函数的完备黎曼流形的例子.在此基础上,证明了Chen-Zheng-Yang的一个猜想是成立的.

We study the eigenvalue problem of the fourth-order elliptic operators in divergence form on complete Riemannian manifolds and obtain a general inequality. By using this general inequaity, we prove universal inequalities for eigenvalues of the fourth-order elliptic operators in divergence form on complete Riemannian manifolds admitting special functions and give some examples for these Riemannian manifolds admitting special functions. Moreover, we solve Chen-Zheng-Yang＇s conjecture in these complete Riemannian manifolds admitting special functions.