A second eigenvalue bound for the Dirichlet Laplacian in hyperbolic space (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
Sharp bounds for higher Steklov-Dirichlet eigenvalues on domains with spherical holes
arXiv (Cornell University), 2023
We consider mixed Steklov-Dirichlet eigenvalue problem on smooth bounded domains in Riemannian manifolds. Under certain symmetry assumptions on multiconnected domains in R n with a spherical hole, we obtain isoperimetric inequalities for k-th Steklov-Dirichlet eigenvalues for 2 ≤ k ≤ n + 1. We extend Theorem 3.1 of [5] from Euclidean domains to domains in space forms, that is, we obtain sharp lower and upper bounds of the first Steklov-Dirichlet eigenvalue on bounded star-shaped domains in the unit n-sphere and in the hyperbolic space.
Asymptotic Estimates for the Principal Eigenvalue of the Laplacian In a Geodesic Ball
Applied Mathematics and Optimization, 1983
Let M be a compact Riemannian manifold and let B e be a geodesic ball of radius e with center o ~ M. We investigate the asymptotic behavior of)~e, the principal eigenvalue of the Laplace-Beltrami operator on M \ /~ e with homogeneous Dirichlet b o u n d a r y conditions. We prove that ?~e-Cq~,(e) where n = d i m M , q~2(e) = (l o g e-l)-1 and d?n(e)= (n-2) e n 2 (n > 2). In the case where M is a model the constant C is explicitly evaluated.
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.
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 ⋆ ).
On the Faber–Krahn inequality for the Dirichletp-Laplacian
ESAIM: Control, Optimisation and Calculus of Variations, 2014
A famous conjecture made by Lord Rayleigh is the following: "The first eigenvalue of the Laplacian on an open domain of given measure with Dirichlet boundary conditions is minimum when the domain is a ball and only when it is a ball". This conjecture was proved simultaneously and independently by Faber [G. Faber, Beweiss dass unter allen homogenen Membranen von gleicher Fläche und gleicher Spannung die kreisförfegige den leifsten Grundton gibt. Sitz. bayer Acad. Wiss. (1923) 169-172] and Krahn [E. Krahn,Über eine von Rayleigh formulierte Minimaleigenschaftdes Kreises. Math. Ann. 94 (1924) 97-100.]. We shall deal with the p-Laplacian version of this theorem.
On conformal spectral gap estimates of the Dirichlet–Laplacian
St. Petersburg Mathematical Journal, 2020
We study spectral stability estimates of the Dirichlet eigenvalues of the Laplacian in non-convex domains Ω ⊂ R 2. With the help of these estimates we obtain asymptotically sharp inequalities of ratios of eigenvalues in the frameworks of the Payne-Pólya-Weinberger inequalities. These estimates are equivalent to spectral gap estimates of the Dirichlet eigenvalues of the Laplacian in non-convex domains in terms of conformal (hyperbolic) geometry.