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
Most of results of Bestvina and Mogilski [Characterizing certain incomplete infinite-di- mensional absolute retracts. Michigan Math. J., 33, 291-313 (1986)] on strong Z-sets in ANR's and absorbing sets is generaliz...Most of results of Bestvina and Mogilski [Characterizing certain incomplete infinite-di- mensional absolute retracts. Michigan Math. J., 33, 291-313 (1986)] on strong Z-sets in ANR's and absorbing sets is generalized to nonseparable case. It is shown that if an ANR X is locally homotopy dense embeddable in infinite-dimensional Hilbert manifolds and w(U) ---- w(X) (where "w"is the topological weight) for each open nonempty subset U of X, then X itself i,~ homotopy dense embeddable in a Hilbert manifold. It is also demonstrated that whenever X is an AR, its weak product W(X, *) ---- {(xn)=l C X : x~ = * for almost all n} is homeomorphic to a pre-Hilbert space E with E EE. An intrinsic characterization of manifolds modelled on such pre-Hilbert spaces is given.展开更多
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.
文摘Most of results of Bestvina and Mogilski [Characterizing certain incomplete infinite-di- mensional absolute retracts. Michigan Math. J., 33, 291-313 (1986)] on strong Z-sets in ANR's and absorbing sets is generalized to nonseparable case. It is shown that if an ANR X is locally homotopy dense embeddable in infinite-dimensional Hilbert manifolds and w(U) ---- w(X) (where "w"is the topological weight) for each open nonempty subset U of X, then X itself i,~ homotopy dense embeddable in a Hilbert manifold. It is also demonstrated that whenever X is an AR, its weak product W(X, *) ---- {(xn)=l C X : x~ = * for almost all n} is homeomorphic to a pre-Hilbert space E with E EE. An intrinsic characterization of manifolds modelled on such pre-Hilbert spaces is given.