Inequalities for eigenvalues of a clamped plate problem

Let be a connected bounded domain in an -dimensional Euclidean space . Assume that

are eigenvalues of a clamped plate problem or an eigenvalue problem for the Dirichlet biharmonic operator:

Then, we give an upper bound of the -th eigenvalue in terms of the first eigenvalues, which is independent of the domain , that is, we prove the following:

Further, a more explicit inequality of eigenvalues is also obtained.

     

A lower bound for eigenvalues of a clamped plate problem

In this paper, we study eigenvalues of a clamped plate problem. We obtain a lower bound for eigenvalues, which gives an important improvement of results due to Levine and Protter.     

Estimates for lower order eigenvalues of a clamped plate problem

For a bounded domain Ω in a complete Riemannian manifold M ⁿ , we study estimates for lower order eigenvalues of a clamped plate problem. We obtain universal inequalities for lower order eigenvalues. We would like to remark that our results are sharp.     

带权Paneitz算子和带权圆盘振动问题的特征值不等式

本文研究光滑度量测度空间上带权Paneitz算子的闭特征值问题和带权圆盘振动问题,给出Euclid空间、单位球面、射影空间和一般Riemann流形的n维紧子流形的权重Paneitz箅子和带权圆盘振动问题的前n个特征值上界估计.进一步地,本文给出带权Ricci曲率有界的紧致度量测度空间上带权圆盘振动问题的第一特征值的下界...     

A finite-difference approximation for the eigenvalues of the clamped plate

Summary A finite-difference scheme is given for the eigenproblem of the clamped plate. The discrete eigenvalues and eigenvectors are shown to converge to the continuous eigenvalues and eigenvectors likeO(h ²) andO (h ² logh 1/2) respectively.This work, supported by the U.S. Department of the Navy under Contract N 00017-62-C-0604.     

Bounding eigenvalues of clamped plates

Zusammenfassung Es wird ein Verfahren zur Berechnung von oberen und unteren Schranken für die Eigenwerte einer eingespannten Platte beliebiger Form dargestellt. Die Methode besteht darin, exakte Partikulärlösungen der Plattengleichung ( ² u=u) zu konstruieren, welche die Randbedingungen des Plattenproblems approximativ erfüllen. Der Abstand zwischen dem Eigenwert der Partikulärlösung und einem Eigenwert des Plattenproblems kann dann mit Hilfe eines Lemmas vonMoler undPayne abgeschätzt werden.Als Beispiele wurden die ersten Eigenwerte der quadratischen Platte sowie einiger elliptischer Platten berechnet. Für eine spezielle elliptische Platte sind die ersten vier Eigenwerte angegeben.     

Inequalities for lower-order eigenvalues of a fourth-order elliptic operator

In this paper, we investigate the Dirichlet weighted eigenvalues problem of a fourth-order elliptic operator with variable coefficients on a bounded domain with smooth boundary in ? ⁿ . We establish some inequalities for lower-order eigenvalues of this problem. In particular, our results contain an inequality for eigenvalues of the biharmonic operator derived by Cheng, Huang, and Wei.     

Inequalities for eigenvalues of the buckling problem of arbitrary order on bounded domains of

We prove universal inequalities for eigenvalues of the buckling problem of arbitrary order on bounded domains in which are independent of the domains.     

Inequalities for generalized eigenvalues of quaternion matrices

Periodica Mathematica Hungarica - Studying eigenvalues of square matrices is a traditional and fundamental direction in linear algebra. Quaternion matrices constitute an important and extensively...     

Critical eigenvalues for a nonlinear problem

We study the application, , where is the supremum of positive s such that the problem admits a solution. Where B ₁ is the unit ball in We show that is a decreasing function, with where is the unique solution of the problem . We also give the explicit solutions of the problem , when and show that . We show that the problem doesnt admit a solution. In the end, we give a numerical approximation of , when .     

Inequalities for the eigenvalues of the Riesz potential

It is proved that, of all the domains with identical measure, it is the ball that maximizes the first eigenvalue of the Riesz potential. It is shown that the sum of the squares of all the eigenvalues is also maximized in the ball among all the domains with identical measure.     

Inequalities for eigenvalues of homogeneous boundary value problems

Sunto Il limite inferiore classico per i valori propri di una equazione differenziale lineare del secondo ordine nel caso dei punti coniugati è un caso speciale di disuguaglianze per problemi auto-aggiunti generali. Per equazioni auto-aggiunte a coefficiente positivo e parametro moltiplicativo possiamo determinare il segno dei valori propri ed otteniamo una serie completa di limiti inferiori, con esempi nei quali i limiti sono raggiunti. Si mostra che, senza ipotesi speciali sul coefficiente, esistono limiti superiori soltanto per qualche caso del vaiore proprw d'ordine zero. Anche questi limiti sono raggiunti in esempi.

---

---

Dedicated to ProfessorBeniamino Segre

---

Entrata in Redazione il 12 Febbraio 1973.

---

Research partially supported by NSF Grant GP-27960.     

Inequalities for eigenvalues of the polyharmonic operator Δ p

In this paper we consider the bounds of the eigenvalues for a class of polyharmonic operators and obtain the bounds for (n+1)th eigenvalue interm of the firstn eigenvalues. Those estimates do not depend on the domain in which the problem is considered.     

On the isoperimetric inequality for the buckling of a clamped plate

In two dimensions, we prove the existence of a domain minimizing the buckling load of a clamped plate among all domains of given measure. In order to prove that the optimal set is the disk, we discuss possible ways of applying an idea of Willms and Weinberger if the regularity of the optimal set is not a priori assumed.     

A control problem for clamped extensible beams

An optimal control problem for a nonlinear partial integro-differential equation is presented. Existence of an optimal control is obtained, and a minimum principle is proved     

Inequalities for eigenvalues of the drifting Laplacian on Riemannian manifolds

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-Pólya-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.