Upper bounds for Steklov eigenvalues on surfaces

We give explicit isoperimetric upper bounds for all Steklov eigenvalues of a compact orientable surface with boundary, in terms of the genus, the length of the boundary, and the number of boundary components. Our estimates generalize a recent result of Fraser-Schoen, as well as the classical inequalites obtained by Hersch-Payne-Schiffer, whose approach is used in the present paper.

1. Introduction 1.1. Steklov spectrum. Let Σ be a compact orientable surface with boundary, and let ∆ be the Laplace-Beltrami operator associated with a Riemannian metric on Σ. The Steklov eigenvalue problem on Σ is given by: where ∂ n denotes the outward normal derivative. The spectrum of the Steklov problem is discrete and its eigenvalues form a sequence 0 = σ 0 < σ 1 ≤ σ 2 ≤ σ 3 ≤ · · · ր ∞, where each eigenvalue is repeated according to its multiplicity [2]. The eigenfunctions φ k , k = 0, 1, 2 . . . can be chosen to form an orthogonal basis of L 2 (∂Σ). Note that the eigenfunction φ 0 corresponding to σ 0 = 0 is constant.
The Steklov eigenvalues coincide with the eigenvalues of the Dirichletto-Neumann map Λ. If the boundary ∂Σ is smooth, it is a pseudodifferential elliptic operator Λ : C ∞ (∂Σ) → C ∞ (∂Σ) of order one [20], defined by where Hf is the harmonic extension of f to the interior of Σ (i.e. ∆(Hf ) = 0 on Σ). The Dirichlet-to-Neumann map has important applications to inverse problems [6,19].
1.2. Main results. Isoperimetric inequalities for Steklov eigenvalues have been actively studied for more than fifty years [21,22,3,15,12]. In particular, a number of recent papers are concerned with the Steklov spectrum on manifolds with boundary [9,10,14,7]. The following estimate on the first nontrivial Steklov eigenvalue on a surface with boundary was proved by Fraser and Schoen [10]: Here L is the length of the boundary, γ is the genus of the surface and l the number of boundary components. For simply connected planar domains, inequality (1.1) is sharp and was proved by Weinstock in [21]. The goal of this note is to generalize (1.1) to higher eigenvalues. We prove Theorem 1.2. Let Σ be a compact orientable surface of genus γ, such that the boundary ∂Σ has l connected components of total length L. Then for any integer k ≥ 1.
In fact, Theorem 1.2 is a special case (set p = q below) of the following result: for any pair of integers p, q ≥ 1.
1.3. Discussion. It follows from Weyl's law for eigenvalues of the Dirichlet-to-Neumann operator that the linear dependence on k in (1.3) is optimal. For simply connected planar domains, the inequalities (1.5) were obtained by Hersch, Payne, and Schiffer in [16]. In [12] we proved that in this case (here γ = 0, l = 1) the estimates (1.3) are sharp for all k ≥ 1. We do not expect (1.3) to be sharp for other values of γ and l (cf. [10, Theorem 2.5]); see also Question 1.8 below.
The proof of Theorem 1.4 combines the methods of [10] and [16]. Following [10], we use a version of Ahlfors Theorem [1] proved by Gabard [11], according to which any Riemannian surface of genus γ with l boundary components can be represented as a proper conformal branched cover of a disk D with degree at most γ + l. Properness of the covering map implies that the boundary ∂Σ is mapped to the circle S 1 . It is essential in this proof since the test functions for the variational characterization of the eigenvalues σ k are built from the eigenfunctions of a certain one-dimensional problem on S 1 . This approach was suggested by Hersch, Payne and Schiffer in [16].
The analogue of the estimate (1.1) for the first nonzero Laplace eigenvalue λ 1 on a closed surface Σ (without boundary) is the Yang-Yau inequality [23] : Note that the bound (1.7) does not depend on the number of boundary components of ∂Σ, which makes it a sharper estimate than (1.3) for l large enough. Another interesting development of Korevaar's method for both Laplace and Steklov eigenvalues can be found in [14] where λ k and σ k are bounded by a linear combination of k and γ (instead of its product). However, the constants in [14] are also implicit.
Let us conclude by an open question. It was proved in [5] that there exists a sequence of closed surfaces Σ n of genera γ n → ∞ such that Moreover, it was subsequently shown in [4] that one can choose a sequence of surfaces with γ n = n and λ 1 (Σ n ) Area(Σ n ) growing linearly as n ր ∞. Therefore, the dependence on the genus γ in the Yang-Yau inequality (1.6) is optimal up to a multiplicative constant. (2. 2) The proof of Theorem 1.4 is based on the approach of [16]. We construct test functions using linear combinations of harmonic oscillators on S 1 , extend them harmonically to the disk and then lift to a branched cover representation of Σ. Using sufficiently many harmonic oscillators, one can ensure the existence of a linear combination satisfying the orthogonality conditions (2.1).
As was shown in [11], there exists a proper conformal branched cover ψ : Σ → D of degree d ≤ γ + l. Because ψ is proper, it takes the boundary ∂Σ to the circle S 1 = ∂D. The restriction of ψ to each connected component of ∂Σ is a covering map of S 1 . Let ds be the Riemannian measure on ∂Σ. We define the push-forward measure dµ = ψ * ds on the circle S 1 , and introduce the "mass parameter" In particular, dµ = m ′ (θ)dθ is absolutely continuous with respect to the Lebesgue measure dθ, and the length of the boundary ∂Σ is given by Given a smooth periodic function h : R → R with period L, define f : h(m(θ)).
The function f admits a unique harmonic extension u to the disk D.
Because the disk is simply connected, this function has a unique harmonic conjugate v normalized in such a way that Let the functions α, β : Σ → R be defined by Recall that the map ψ is a d-fold conformal branched covering of D. It follows from conformal invariance of the Dirichlet energy in two dimensions (see also [23]) that Moreover, the Cauchy-Riemann equations imply that these two quantities are equal. Integration by parts gives Multiplying the two equations in (2.4) and using (2.5), we get The Cauchy-Riemann equations also imply the pointwise equality Applying the Cauchy-Schwarz inequality to the measure dµ = m ′ (θ)dθ leads to : At the same time, Estimating the product of the Rayleigh quotients R α := R Σ (α) and R β := R Σ (β) using the relations (2.6), (2.7) and (2.8), we notice that S 1 v 2 (θ) dµ(θ) cancels out on the right-hand side. This is the key trick in the method introduced in [16]. Namely, we obtain the following bound: Here dm is the Rayleigh quotient of a uniform circular string of length L. Its eigenmodes are well known. Let h k : R → R, k = 0, 1, 2 . . . , be defined by h 0 = 1 and for k ≥ 1. Clearly, This leads to 2.2. Construction of test-functions. The rest of the argument is almost exactly the same as in [16]. We present it below for the sake of completeness. Let N = p + q − 1. Consider a function where the functions f k : S 1 → R are defined by f k (θ) = h k (m(θ). The functions f k are dµ-orthogonal to each other, and hence linearly independent. The harmonic extensions u k of f k are also linearly independent, because taking the harmonic extension is a linear and injective operation. For the same reason, the harmonic conjugates v k are linearly independent as well. Moreover, since by definition f 0 = 1, f k are dµorthogonal to constants for all k = 1, 2, 3, . . . , and hence ∂Σ α k = 0 for all k ≥ 1, where α k = u k • ψ. At the same time, by the normalization As before, these functions are lifted to α = u • ψ and β = v • ψ.
In order to use u and v in the variational characterization (2.2) for σ p and σ q respectiveley, they have to satisfy the orthogonality conditions (2.1) : ∂Σ αφ k = 0 for k = 1, · · · , p − 1 ∂Σ βφ k = 0 for k = 1, · · · , q − 1 These N − 1 linear constraints can be resolved for some choice of N constants c 1 , . . . , c N . It follows from (2.9) that where h = N k=1 c k h k . We conclude by observing that Recalling that d ≤ γ + l completes the proof of Theorem 1.4.