In this paper, we study eigenvalues of the poly-Laplacian with arbitrary order on a bounded domain in an n-dimensional Euclidean space and obtain a lower bound for eigenvalues, which gives an important improvement of results due to Levine and Protter [12]. In particular, the result of Melas [15] is included here.2010 Mathematics Subject Classification: 35P15. Key words and phrases: the eigenvalue problem, a lower bound for eigenvalues, the ploy-Laplacian with arbitrary order.
This paper studies eigenvalues of the buckling problem of arbitrary order on bounded domains in Euclidean spaces and spheres. We prove universal bounds for the k-th eigenvalue in terms of the lower ones independent of the domains. Our results strengthen the recent work in [28] and generalize Cheng-Yang's recent estimates [16] on the buckling eigenvalues of order two to arbitrary order.2000 Mathematics Subject Classification : 35P15, 53C20, 53C42, 58G25 Key words and phrases: Universal inequality for eigenvalues, the buckling problem of arbitrary order, Euclidean space, sphere.
For a bounded domain Ω with a piecewise smooth boundary in a complete Riemannian manifold M, we study eigenvalues of the Dirichlet eigenvalue problem of the Laplacian. By making use of a fact that eigenfunctions form an orthonormal basis of L2(Ω) in place of the Rayleigh–Ritz formula, we obtain inequalities for eigenvalues of the Laplacian. In particular, for lower order eigenvalues, our results extend the results of Chen and Cheng [D. Chen and Q.-M. Cheng, Extrinsic estimates for eigenvalues of the Laplace operator, J. Math. Soc. Japan 60 (2008) 325–339].
Let x : M → R N be an n-dimensional compact self-shrinker in R N with smooth boundary ∂Ω. In this paper, we study eigenvalues of the operator Lr on M , where Lr is defined by Lr = e |x| 2 2 div(e − |x| 2 2 T r ∇·) with T r denoting a positive definite (0,2)-tensor field on M . We obtain "universal" inequalities for eigenvalues of the operator Lr. These inequalities generalize the result of Cheng and Peng in [8]. Furthermore, we also consider the case that equalities occur.2000 Mathematics Subject Classification. Primary 53C40, Secondary 58C40.
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