L_(ν) operator is an important extrinsic differential operator of divergence type and has profound geometric settings.In this paper,we consider the clamped plate problem of L_(ν)^(2)operator on a bounded domain of t...L_(ν) operator is an important extrinsic differential operator of divergence type and has profound geometric settings.In this paper,we consider the clamped plate problem of L_(ν)^(2)operator on a bounded domain of the complete Riemannian manifolds.A general formula of eigenvalues of L_(ν)^(2) operator is established.Applying this general formula,we obtain some estimates for the eigenvalues with higher order on the complete Riemannian manifolds.As several fascinating applications,we discuss this eigenvalue problem on the complete translating solitons,minimal submanifolds on the Euclidean space,submanifolds on the unit sphere and projective spaces.In particular,we get a universal inequality with respect to the L_(II) operator on the translating solitons.Usually,it is very difficult to get universal inequalities for weighted Laplacian and even Laplacian on the complete Riemannian manifolds.Therefore,this work can be viewed as a new contribution to universal estimate.展开更多
In this paper, we first establish an abstract inequality for lower order eigenvalues of a self-adjoint operator on a Hilbert space which generalizes and extends the recent results of Cheng et al. （Calc. Var. Partial ...In this paper, we first establish an abstract inequality for lower order eigenvalues of a self-adjoint operator on a Hilbert space which generalizes and extends the recent results of Cheng et al. （Calc. Var. Partial Differential Equations, 38, 409-416 （2010））. Then, making use of it, we obtain some universal inequalities for lower order eigenvalues of the biharmonic operator on manifolds admitting some speciM functions. Moreover, we derive a universal inequality for lower order eigenvalues of the poly-Laplacian with any order on the Euclidean space.展开更多
基金Supported by the National Natural Science Foundation of China(Grant Nos.11861036 and 11826213)the Natural Science Foundation of Jiangxi Province(Grant No.20224BAB201002)。
文摘L_(ν) operator is an important extrinsic differential operator of divergence type and has profound geometric settings.In this paper,we consider the clamped plate problem of L_(ν)^(2)operator on a bounded domain of the complete Riemannian manifolds.A general formula of eigenvalues of L_(ν)^(2) operator is established.Applying this general formula,we obtain some estimates for the eigenvalues with higher order on the complete Riemannian manifolds.As several fascinating applications,we discuss this eigenvalue problem on the complete translating solitons,minimal submanifolds on the Euclidean space,submanifolds on the unit sphere and projective spaces.In particular,we get a universal inequality with respect to the L_(II) operator on the translating solitons.Usually,it is very difficult to get universal inequalities for weighted Laplacian and even Laplacian on the complete Riemannian manifolds.Therefore,this work can be viewed as a new contribution to universal estimate.
基金supported by National Natural Science Foundation of China(Grant No.11001130)
文摘In this paper, we first establish an abstract inequality for lower order eigenvalues of a self-adjoint operator on a Hilbert space which generalizes and extends the recent results of Cheng et al. （Calc. Var. Partial Differential Equations, 38, 409-416 （2010））. Then, making use of it, we obtain some universal inequalities for lower order eigenvalues of the biharmonic operator on manifolds admitting some speciM functions. Moreover, we derive a universal inequality for lower order eigenvalues of the poly-Laplacian with any order on the Euclidean space.
