Universal Bounds for Eigenvalues of a Buckling Problem
TL;DR: In this paper, a new method to construct ''nice'' trial functions was introduced and a universal inequality for higher eigenvalues of the buckling problem was derived by making use of the trial functions.
Abstract: In this paper, we investigate an eigenvalue problem for a biharmonic operator on a bounded domain in an n-dimensional Euclidean space, which is also called a buckling problem. We introduce a new method to construct ``nice'' trial functions and we derive a universal inequality for higher eigenvalues of the buckling problem by making use of the trial functions. Thus, we give an affirmative answer for the problem on universal bounds for eigenvalues of the buckling problem, which was proposed by Payne, Polya and Weinberger in [14] and this problem has been mentioned again by Ashbaugh in [1].
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TL;DR: In this article, it was shown that the perturbed Krein Laplacian (i.e., the Krein-von Neumann extension of − Δ + V defined on C 0 ∞ ( Ω ) is spectrally equivalent to the buckling of a clamped plate problem.
78 citations
TL;DR: In this article, the eigenvalues of a clamped plate problem on complete manifolds are studied and a universal inequality for the case of warped product manifolds is shown. But the universal inequalities are not applicable to the complete manifold with Ricci curvatures.
Abstract: In this paper we study eigenvalues of a clamped plate problem on compact domains in complete manifolds. For complete manifolds admitting special functions, we prove universal inequalities for eigenvalues of clamped plate problem independent of the domains of Payne–Polya–Weinberger–Yang type. These manifolds include Hadamard manifolds with Ricci curvature bounded below, a class of warped product manifolds, the product of Euclidean spaces with any complete manifolds and manifolds admitting eigenmaps to a sphere. In the case of warped product manifolds, our result implies a universal inequality on hyperbolic space proved by Cheng–Yang. We also strengthen an inequality for eigenvalues of clamped plate problem on submanifolds in a Euclidean space obtained recently by Cheng, Ichikawa and Mametsuka.
70 citations
TL;DR: Li and Yau as mentioned in this paper showed that λ k ⩾ 4 π 2 ( ω n vol Ω ) 2 n k 2 n, for k = 1, 2, …, which is sharp in the sense of average.
47 citations
TL;DR: In this paper, the eigenvalues of the drifting Laplacian on compact Riemannian manifolds with boundary (possibly empty) were studied and a general inequality for them was derived.
Abstract: This paper studies eigenvalues of the drifting Laplacian on compact Riemannian manifolds with boundary (possibly empty) and provides a general inequality for them. Using the general inequality, we obtain universal inequalities for eigenvalues of the drifting Laplacian of Payne-Polya-Weinberger-Yang type for manifolds supporting some special functions. We also obtain a lower bound for the first eigenvalue of the square of the drifting Laplacian on compact manifolds with boundary under some condition on the Bakry-Ricci curvature.
45 citations
TL;DR: In this paper, the authors proved the unitary equivalence of the inverse of the Krein-von Neumann extension (on the orthogonal complement of its kernel) of a densely defined, closed, strictly positive operator, ≥ e ℋ for some e > 0 in a Hilbert space 210B to an abstract buckling operator.
Abstract: In the first (and abstract) part of this survey we prove the unitary equivalence of the inverse of the Krein–von Neumann extension (on the orthogonal complement of its kernel) of a densely defined, closed, strictly positive operator, ≥ e ℋ for some e > 0 in a Hilbert space 210B to an abstract bucklingpr oblem operator.
42 citations
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311 citations
TL;DR: In this paper, a trace identity for the spectra of self-adjoint operators H modeled on the Dirichlet Laplacian or, more generally, on Schrodinger operators of the form (p−A(x))2 + V (x), where p = 1i ∇ is the usual momentum operator in convenient units and A(x) is the magnetic vector potential.
Abstract: In this article, we prove and exploit a trace identity for the spectra of Schrodinger operators and similar operators. This identity leads to universal bounds on the spectra, which apply to low-lying eigenvalues, eigenvalue asymptotics, and to partition functions (traces of heat operators). In many cases they are sharp in the sense that there are specific examples for which the inequalities are saturated. Special cases corresponding to known inequalities include those of Hile and Protter and of Yang. Introduction In this article, we prove and exploit an identity for the spectra of self-adjoint operators H modeled on the Dirichlet Laplacian or, more generally, on Schrodinger operators of the form (p−A(x))2 + V (x), (1) where p = 1i ∇ is the usual momentum operator in convenient units and A(x) is the magnetic vector potential. We recover and extend several known inequalities involving sums, differences, and ratios of eigenvalues. Let λj , j = 1, . . . , denote the ordered eigenvalues of the Dirichlet Laplacian on a bounded d-dimensional domain with zero Dirichlet boundary conditions, and recall that Hile and Protter [HiPr80] proved that: d 4 ≤ 1 n n ∑ j=1 λj λn+1 − λj , (2) thereby extending an earlier inequality of Payne, Polya, and Weinberger [PaPoWe56]. In the last few years it has become clear that these and many similar relationships can be realized as special cases of abstract variational bounds involving the interplay among commutators of −∇2, a Cartesian coordinate xj , and the corresponding derivative ∂/∂xj. For this analysis and various extensions of (2) see [Ha88], [Ho90], [Ha93], [HaMi95]. While inequalities of this type have been fairly sharp for low-lying eigenvalues, they have been mostly disappointing for higher eigenvalues. Yang [Ya91], however, Received by the editors September 28, 1995. 1991 Mathematics Subject Classification. Primary 35J10, 35J25, 58G25.
112 citations
TL;DR: In this article, the eigenvalues of Laplacian on a bounded connected domain in an n-dimensional unit sphere Sn(1) or a compact homogeneous Riemannian manifold were studied.
Abstract: In this paper, we study eigenvalues of Laplacian on either a bounded connected domain in an n-dimensional unit sphere Sn(1), or a compact homogeneous Riemannian manifold, or an n-dimensional compact minimal submanifold in an N-dimensional unit sphere SN(1). We estimate the k+1-th eigenvalue by the first k eigenvalues. As a corollary, we obtain an estimate of difference between consecutive eigenvlaues. Our results are sharper than ones of P. C. Yang and Yau [25], Leung [19], Li [20] and Harrel II and Stubbe [12], respectively. From Weyl’s asymptotical formula, we know that our estimates are optimal in the sense of the order of k for eigenvalues of Laplacian on a bounded connected domain in an n-dimensional unit sphere Sn(1).
109 citations
99 citations
TL;DR: In this article, an upper bound of the (k+1)-th eigenvalue λ k+1 in terms of the first k eigenvalues, which is independent of the domain D, is obtained.
Abstract: Let D be a connected bounded domain in an n-dimensional Euclidean space R n . Assume that 0 < λ 1 < λ 2 ≤ ··· ≤λ k ≤··· are eigenvalues of a clamped plate problem or an eigenvalue problem for the Dirichlet biharmonic operator: Then, we give an upper bound of the (k+1)-th eigenvalue λ k+1 in terms of the first k eigenvalues, which is independent of the domain D, that is, we prove the following: Further, a more explicit inequality of eigenvalues is also obtained.
96 citations