Estimates for eigenvalues of Neumann and Navier problem
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In this paper, we firstly prove some general inequalities for the Neumann eigenvalues for domains contained in a Euclidean $n$-space $\R^n$. Using the general inequalities, we can derive some new Neumann eigenvalues estimates which include an upper bound for the $(k+1)^{th}$ eigenvalue and a new estimate for the gap of the consecutive eigenvalues. Moreover, we give sharp lower bound for the first eigenvalue of two kinds of eigenvalue problems of the biharmonic operator with Navier boundary condition on compact Riemannian manifolds with boundary and positive Ricci curvature.
Abstract
In this paper, we firstly prove some general inequalities for the Neumann eigenvalues for domains contained in a Euclidean $n$-space $\R^n$. Using the general inequalities, we can derive some new Neumann eigenvalues estimates which include an upper bound for the $(k+1)^{th}$ eigenvalue and a new estimate for the gap of the consecutive eigenvalues. Moreover, we give sharp lower bound for the first eigenvalue of two kinds of eigenvalue problems of the biharmonic operator with Navier boundary condition on compact Riemannian manifolds with boundary and positive Ricci curvature.
참고문헌 (11)
- Daguang Chen, "On eigenvalues of a system of elliptic equations and of the biharmonic operator", Journal of Mathematical Analysis and Applications, 387(2), pp.1146~1159 (2012), Elsevier BV[DOI]
- F.R.K. Chung, "Upper Bounds for Eigenvalues of the Discrete and Continuous Laplace Operators", Advances in Mathematics, 117(2), pp.165~178 (1996), Elsevier BV[DOI]
- E. M. Harrell, II, "Commutator bounds for eigenvalues of some differential operators, in Evolution equations", pp.235~244 (1995), Dekker
- G. Hile, "Inequalities for eigenvalues of the Laplacian", Indiana University Mathematics Journal, 29(4), pp.523~538 (1980), Indiana University Mathematics Journal[DOI]
- Saïd Ilias, "Universal inequalities for the eigenvalues of a power of the Laplace operator", manuscripta mathematica, 132(1-2), pp.75~102 (2010), Springer Science and Business Media LLC[DOI]
- Michael Levitin, "Commutators, Spectral Trace Identities, and Universal Estimates for Eigenvalues", Journal of Functional Analysis, 192(2), pp.425~445 (2002), Elsevier BV[DOI]
- L. E. Payne, "On the Ratio of Consecutive Eigenvalues", Journal of Mathematics and Physics, 35(1-4), pp.289~298 (1956), Wiley[DOI]
- Robert Reilly, "Applications of the Hessian operator in a Riemannian manifold", Indiana University Mathematics Journal, 26(3), pp.459~472 (1977), Indiana University Mathematics Journal[DOI]
- Colin J. Thompson, "On the Ratio of Consecutive Eigenvalues inN-Dimensions", Studies in Applied Mathematics, 48(3), pp.281~283 (1969), Wiley[DOI]
- Q. Wang, "Sharp lower bounds for the first eigenvalues of the bi-Laplace operator", arXiv : 1802. 05502v5, ~ (2018)
- H. C. Yang, "An estimate of the difference between consecutive eigenvalues, preprint IC", 91, pp.1991, ~