A sharp bound for the ratio of the first two Dirichlet eigenvalues of a domain in a hemisphere of 𝕊ⁿ

Abstract

Abstract. 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.

Spectral Problems in Spaces of Constant Curvature

“…Then, one has to prove monotonicity properties for functions associated to the trial function; iv) then one uses symmetrization (symmetric decreasing rearrangements), and, finally, v) a comparison theorem due to G. Chiti. The analogous result for domains in S n was proved in [9], but, again, only for domains contained in a hemisphere. It is still an open problem to determine whether a PPW inequality, like (5) is true for domains in S n extending beyond a hemisphere.…”

“…The main reason (but not the only one) that our proof fails for domain that extends beyond a hemisphere is that we cannot prove the monotonicity properties associated to the trial functions (i.e., the main part of step iii) in the proof of PPW). This failure relies on the fact that the Laplace-Beltrami operator (in geodesic ccordinates) acting on radial functions has a term proportional to the radial derivative and the coefficient changes sign when going beyond the hemisphere (see equation (9) below. On the contrary, for both the Euclidean and the Hyperbolic cases the corresponding coefficient is of one sign.…”

Isoperimetric and universal inequalities for eigenvalues

“…On the other hand, the proof found in [13] (see also [16]) is somewhat simpler and lends itself to generalization to a version of the PPW conjecture for domains in S n . These topics are dealt with further in [20], [21], and also to some extent below.…”

On Ashbaugh-Benguria's Conjecture about Lower Order Dirichlet Eigenvalues of the Laplacian

“…Finally, Ashbaugh and Benguria proved this conjecture [2,3,4]. Ashbaugh-Benguria [7] and Benguria-Linde [8] also proved similar inequalities for the first two Dirichlet eigenvalues of the Laplacian on bounded domains in a hemisphere and a hyperbolic space, respectively.…”