A Sharp Estimate for Neumann Eigenvalues of the Laplace-Beltrami Operator for Domains in a Hemisphere (original) (raw)
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2001
For a domain Ω contained in a hemisphere of the n-dimensional sphere S n we prove the optimal result λ 2 /λ 1 (Ω) ≤ λ 2 /λ 1 (Ω ) for the ratio of its first two Dirichlet eigenvalues where Ω , the symmetric rearrangement of Ω in S n , is a geodesic ball in S n having the same n-volume as Ω. We also show that λ 2 /λ 1 for geodesic balls of geodesic radius θ 1 less than or equal to π/2 is an increasing function of θ 1 which runs between the value (j n/2,1 /j n/2−1,1 ) 2 for θ 1 = 0 (this is the Euclidean value) and 2(n + 1)/n for θ 1 = π/2. Here j ν,k denotes the kth positive zero of the Bessel function Jν (t). This result generalizes the Payne-Pólya-Weinberger conjecture, which applies to bounded domains in Euclidean space and which we had proved earlier. Our method makes use of symmetric rearrangement of functions and various technical properties of special functions. We also prove that among all domains contained in a hemisphere of S n and having a fixed value of λ 1 the one with the maximal value of λ 2 is the geodesic ball of the appropriate radius. This is a stronger, but slightly less accessible, isoperimetric result than that for λ 2 /λ 1 . Various other results for λ 1 and λ 2 of geodesic balls in S n are proved in the course of our work.
2000
For a domain Omega\OmegaOmega contained in a hemisphere of the nnn-dimensional sphere SSn\SS^nSSn we prove the optimal result lambda2/lambda1(Omega)lelambda2/lambda1(Omegastar)\lambda_2/\lambda_1(\Omega) \le \lambda_2/\lambda_1(\Omega^{\star})lambda_2/lambda_1(Omega)lelambda2/lambda1(Omegastar) for the ratio of its first two Dirichlet eigenvalues where Omegastar\Omega^{\star}Omegastar, the symmetric rearrangement of Omega\OmegaOmega in SSn\SS^nSSn, is a geodesic ball in SSn\SS^nSSn having the same nnn-volume as Omega\OmegaOmega. We also show that lambda2/lambda1\lambda_2/\lambda_1lambda_2/lambda_1 for geodesic balls of
Two-term, asymptotically sharp estimates for eigenvalue means of the Laplacian
Journal of Spectral Theory
We present asymptotically sharp inequalities for the eigenvalues µ k of the Laplacian on a domain with Neumann boundary conditions, using the averaged variational principle introduced in [14]. For the Riesz mean R1(z) of the eigenvalues we improve the known sharp semiclassical bound in terms of the volume of the domain with a second term with the best possible expected power of z. In addition, we obtain two-sided bounds for individual µ k , which are semiclassically sharp. In a final section, we remark upon the Dirichlet case with the same methods.
Berezin–Li–Yau inequalities on domains on the sphere
Journal of Mathematical Analysis and Applications, 2019
We prove Berezin-Li-Yau inequalities for the Dirichlet and Neumann eigenvalues on domains on the sphere S d−1. A sharp explicit bound for the sums of the Neumann eigenvalues is obtained for all dimensions d. In the case of S 2 we also obtain sharp lower bounds with correction terms for the vector Laplacian and the Stokes operator.
A Second Eigenvalue Bound for the Dirichlet Schrödinger Operator
Communications in Mathematical Physics, 2006
Let λ i (Ω, V ) be the ith eigenvalue of the Schrödinger operator with Dirichlet boundary conditions on a bounded domain Ω ⊂ R n and with the positive potential V . Following the spirit of the Payne-Pólya-Weinberger conjecture and under some convexity assumptions on the spherically rearranged potential V ⋆ , we prove that λ 2 (Ω, V ) ≤ λ 2 (S 1 , V ⋆ ). Here S 1 denotes the ball, centered at the origin, that satisfies the condition λ 1 (Ω, V ) = λ 1 (S 1 , V ⋆ ).
About Bounds for Eigenvalues of the Laplacian with Density
Symmetry, Integrability and Geometry: Methods and Applications, 2020
Let M denote a compact, connected Riemannian manifold of dimension n ∈ N. We assume that M has a smooth and connected boundary. Denote by g and dv g respectively, the Riemannian metric on M and the associated volume element. Let ∆ be the Laplace operator on M equipped with the weighted volume form dm := e −h dv g. We are interested in the operator L h • := e −h(α−1) (∆ • +αg(∇h, ∇•)), where α > 1 and h ∈ C 2 (M) are given. The main result in this paper states about the existence of upper bounds for the eigenvalues of the weighted Laplacian L h with the Neumann boundary condition if the boundary is non-empty.
Isoperimetric inequalities for eigenvalues of the Laplacian
Contemporary Mathematics, 2011
This paper deals with eigenvalue optimization problems for a family of natural Schrödinger operators arising in some geometrical or physical contexts. These operators, whose potentials are quadratic in curvature, are considered on closed surfaces immersed in space forms and we look for geometries that maximize the eigenvalues. We show that under suitable assumptions on the potential, the first and the second eigenvalues are maximized by (round) spheres.
A second eigenvalue bound for the Dirichlet Schroedinger operator
2005
Let λ_i(Ω,V) be the ith eigenvalue of the Schrödinger operator with Dirichlet boundary conditions on a bounded domain Ω⊂^n and with the positive potential V. Following the spirit of the Payne-Pólya-Weinberger conjecture and under some convexity assumptions on the spherically rearranged potential V_, we prove that λ_2(Ω,V) <λ_2(S_1,V_). Here S_1 denotes the ball, centered at the origin, that satisfies the condition λ_1(Ω,V) = λ_1(S_1,V_). Further we prove under the same convexity assumptions on a spherically symmetric potential V, that λ_2(B_R, V) / λ_1(B_R, V) decreases when the radius R of the ball B_R increases. We conclude with several results about the first two eigenvalues of the Laplace operator with respect to a measure of Gaussian or inverted Gaussian density.
Isoperimetric bounds for Wentzel-Laplace eigenvalues on Riemannian manifolds
Cornell University - arXiv, 2020
In this paper, we investigate eigenvalues of the Wentzel-Lapla ce operator on a bounded domain in some Riemannian manifold. We prove asymptotically optimal estimates, according to the Weyl's law through bounds that are given in terms of the isoperimetric ratio of the domain. Our results show that the isoperimetric ratio allows to control the entire spectrum of the Wentzel-Laplace operator in various ambient spaces. 1 + c n Vol(Ω)∆B Vol(B) 2 , c n := Vol(Ω) n−1 n. In the numerator Vol stands for the (n − 1)-Riemannian volume and for the n-Riemannian volume from g in the denominator. Our first result provides an upper bound in the case of Euclidean domains. We respectively denote ω n and ρ n−1 = nω n the volumes of the unit ball and the unit sphere in the n-dimensional Euclidean space. Theorem 1.1. Let n 3 and Ω ⊂ IR n be a bounded euclidean domain with smooth boundary Γ. Then, for every k 1, one has λ β W,k (Ω) ζ 1 (n) Vol(Ω) Vol(Γ) 1− 2 n k Vol(Γ) 2 n + ζ 2 (n)I(Ω) 1+ 2 n−1