An isoperimetric inequality for the harmonic mean of the Steklov eigenvalues in hyperbolic space
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Archiv der Mathematik
An isoperimetric inequality for the harmonic mean of the Steklov eigenvalues in hyperbolic space Sheela Verma
Abstract. In this article, we prove an isoperimetric inequality for the harmonic mean of the first (n − 1) nonzero Steklov eigenvalues on bounded domains in n-dimensional hyperbolic space. Our approach to prove this result also gives a similar inequality for the first n nonzero Steklov eigenvalues on bounded domains in n-dimensional Euclidean space. Mathematics Subject Classification. Primary 35P15; Secondary 58J50. Keywords. Isoperimetric inequality, Steklov eigenvalue problem, Exponential map, Geodesic normal coordinate system.
1. Introduction. Let Ω be a bounded domain in a complete Riemannian manifold (M, ds2 ) with smooth boundary ∂Ω. Consider the Steklov eigenvalue problem on Ω Δu = 0 in Ω, ∂u ∂ν = μu on ∂Ω. Here Δ := −div(grad u), ν is the outward unit normal to ∂Ω, and ∂u ∂ν denotes the directional derivative of u in the direction ν. This problem was introduced by Steklov [11] for bounded domains in the plane in 1902. Its importance lies in the fact that the set of eigenvalues of the Steklov problem is the same as the set of eigenvalues of the well known Dirichlet-Neumann map. This map associates to each function u defined on ∂Ω the normal derivative of its harmonic extension on Ω. It is known that the Steklov eigenvalues are discrete and form an increasing sequence 0 = μ0 (Ω) < μ1 (Ω) ≤ μ2 (Ω) ≤ · · · ∞. The interplay between the geometry of the manifold and the Steklov eigenvalues has recently attracted substantial attention. See [3–5,7,9,10] and the references therein for recent developments. The problem of finding a domain
S. Verma
Arch. Math.
under some geometric constraints, which optimizes the eigenvalues (or some combination of eigenvalues), is a classical question in spectral geometry. In this direction, the first result for the Steklov eigenvalues was given by Weinstock [12] in 1954. Using conformal map technique, he proved that among all simply connected planar domains with analytic boundary of fixed perimeter, the circle maximizes μ1 . Hersch and Payne [6] noticed that Weinstock’s proof gives a sharper isoperimetric inequality 1 P (Ω) 1 + ≥ , (1) μ1 (Ω) μ2 (Ω) π where P (Ω) represents the perimeter of Ω ⊂ R2 . Later F. Brock [2] generalized result (1) to Rn by fixing the volume of the domain and proved the following inequality: for a bounded Lipschitz domain Ω ⊂ Rn , n 1 n ≥ , (2) μ (Ω) μ (B(R)) 1 i=1 i where B(R) ⊂ Rn is a ball of radius R such that vol(Ω) = vol(B(R)). In this paper, we extend Brock’s result and prove an isoperimetric inequality for the sum of reciprocals of the first (n − 1) nonzero Steklov eigenvalues on bounded domains in n-dimensional hyperbolic space. Further, our technique also provides a geometric proof of inequality (2). The main results of this article are as follows. Theorem 1.1. Let Hn be the n-dimensional hyperbolic space with constant curvature −1 and Ω ⊂ Hn be a bounded domain with smooth boundary ∂Ω. Then n−1 i=1
n−1 1 1 ≥ ,
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