A Payne-Weinberger eigenvalue estimate for wedge domains on spheres

A Faber-Krahn type argument gives a sharp lower estimate for the first Dirichlet eigenvalue for subdomains of wedge domains in spheres, generalizing the inequality in the plane, found by Payne and Weinberger. An application is an alternative proof to the finiteness of a Brownian motion capture time estimate.

Many lower estimates for the first Dirichlet eigenvalue of a domain stem from an inequality between a line integral and an area integral [Ch,, [LT,, [P, pp. 462-467]. These inequalities are often sharp, in that equality of the eigenvalues implies a geometric equality. For example, the Faber-Krahn inequality [F], [K], proved by comparing level sets of the eigenfunction using the classical isoperimetric inequality, reduces to equality for round disks. Cheeger's inequality [C] bounds the eigenvalue from below in terms of the minimal ratio of area to length of subdomains.
Our main result, Theorem 1, is a lower bound for the first Dirichlet eigenvalue for a domain contained in a wedge in a two sphere, generalizing an eigenvalue estimate of Payne and Weinberger [PW], [P, p.462] for planar domains contained in a wedge. As an application, we give an alternative proof of our Brownian capture time estimate [RT]. Curiously, our proof does not seem to carry over to domains contained in a wedge in the hyperbolic plane.
If (ρ, θ) are polar coordinates centered at a pole of S 2 , recall that the round metric is given by ds 2 = dρ 2 + sin 2 ρ dθ 2 .
Let W = {(ρ, θ) : 0 ≤ θ ≤ π/α, 0 ≤ ρ < π} be the sector in S 2 of angle π/α, for α > 1, and let G be a domain such that G ⊂ W is compact. Also define the truncated sector S(r) is a positive harmonic function in W, with zero boundary values.
Theorem 1. For every subdomain G with compact G ⊂ W, we have the estimate where r * is chosen such that Equality holds if and only if G is the sector S(r * ).
Our argument is similar to the proof of the planar version in [PW]. Our main tool is an isoperimetric-type inequality, Lemma 3, which we prove in Section 1. We use this inequality to estimate the Rayleigh quotient of a test function, proving Theorem 1, in Section 2. Finally, in Section 3, we apply our eigenvalue estimate to a problem in Brownian pursuit.

Isoperimetric Inequality
In this section we prove an isoperimetric inequality for moments of inertia of a domain G ⊂ W. Later we will use this inequality to estimate the Raleigh quotient of admissible functions in G.
We begin by stating a version Szegő's Lemma [Sz]: (3) For φ increasing, equality holds if and only if the measure of E ∩ [0, R] is R.
Proof. Let µ be Lesbesgue measure with line element dx and define the measure ν by dν = ψ dx. Then ν is absolutely continuous with respect to µ and, using the Radon-Nikodym Theorem, when we change variables y = Ψ(x) we have dy = ψ(x)dx. Let E ′ be the image of E under the map Ψ, with characteristic function χ E ′ , so that Φ( E ′ dy) = Φ( E ψ(x)dx). Next, because φ is nondecreasing, for y ≥ 0, Moreover, for φ increasing, equality holds if and only if µ(E ′ ∩ [0, y]) = y. We multiply this inequality by χ E ′ and integrate: On the other hand, Putting these two inequalities together yields the inequality (3).
Lemma 3. Let G ⊂ W be a domain with compact closure. Then there is a function Here F (ρ) = tan 2α (ρ/2) sin ρ and Z is given by (11). Equality holds if and only if G is a sector S(r).
Proof. Map the domain G into a domainG in the upper halfplane using the transformation where we will choose f to satisfy formula (8). The Euclidean line element is We claim that the map satisfies For this to be true pointwise, we need the inequalities to hold Expand sin ρ = 2 sin(ρ/2) cos(ρ/2) and use equality in inequality (7) to define f : Differentiating, we see which implies that the inequality (6) holds as well. Equation (1) and inequality (5) imply that The right side is the moment of inertia of a uniform mass distribution of the curve ∂G relative to the y-axis. Among all domains with given fixed surface moment G y 2 dx dy, the semicircular arcs centered on the y-axis minimize M(∂G) [PW,Section 2]. Compute M(∂G) and M(G) in the case where ∂G is a semicircle of radius R: Solving for R in the formula for M(G) above and using the fact that semicircles are minimizers, we see that for a general domainG in the upper half plane Returning to the original variables, dx dy = αfḟ dρ dθ so Regroup the integral inside the braces Equality holds if and only if H θ = [0, r(θ)] is an interval a.e. Next we let p = 4 3β > 1, q = 4 4−3β , and define the measure dν = sin 2 αθ dθ. Hölder's inequality implies Raising both sides of this inequality to the power p, rearranging, and using the fact that .
We regroup the inside integral again: Let us denote and definer(r, θ) by where χ H denotes the characteristic function of H. The integrand tan 2α (ρ/2) sin ρ is positive and increasing for the range of ρ we are considering, and sor(r, θ) ≤ r with equality if and only if H θ ∩ [0, r] = [0, r] a.e. If we require (2α + 1)β ≥ 2α + 2, then the factor is increasing in ρ. Thus we can define Φ β by Observe that Z and g β are increasing, so φ β is increasing and Φ β is convex. Using g β (r(ρ, θ)) ≤ g β (ρ), we have Now, using Lemma 2 with ψ(ρ) = tan 2α (ρ/2) sin(ρ) χ H θ we have with equality if and only if H θ = [0, r(θ)] is an interval a.e. Next, by Jensen's inequality (with the measure given by dν = sin 2 αθ dθ), with equality if and only ifr(θ) is a.e. constant. Substituting back, Reinserting this back into (9) yields ∂G w 2 ds ≥ π 2α where equality holds if and only if also ρ(θ) is constant a.e. Notice that the right hand side of this inequality is always bounded by ∂G w 2 ds, and so we can use the Dominated Convergence Theorem to take a limit as β → 4 3 from below. In other words, (13) holds for β = 4 3 . Let us compute Φ 1 β β (Y ). Since it depends only on (12), it would be the same for any function v * whose level sets G * η = {x : v * (x) ≥ η} give the same value for the integral of w 2 (see (16) below). In this case, we choose a spherical rearrangement whose levels are the sectors G * η = S(r(η)). Expressing things in terms of r(η), we have so, changing variables s = Z(r) Observe that we get the same equation (13) for all β. Thus we set Υ α = Φ 1 β β in (13) giving (4). It is precisely at inequality (6) where the analagous proof in the hyperbolic case fails. In the hyperbolic case, the harmonic weight function is w(ρ, θ) = tanh 2α (ρ/2) sin(αθ), and versions of equations (5), (8) hold with cos replaced by cosh and sin replaced by sinh. This choice of f gives us f 2ḟ = tanh 2α ρ 2 2α + cosh ρ 3 , much like the formula above, but this does not yield f 2ḟ ≤ α tanh 2α (ρ/2), because cosh ρ grows exponentially with ρ. To rememdy this problem, one can try to vary the power of sinh(ρ/2) or cosh(ρ/2); however this will only yield a worse inequality for f 2ḟ .
2 Estimate of Rayleigh Quotient.
Theorem 1 now follows as in [PW]. Let G ⊂ S 2 be a domain that lies in the wedge W = {(ρ, θ) : 0 ≤ ρ, 0 ≤ θ ≤ π/α}. It suffices to estimate the Rayleigh quotient for admissible functions u ∈ C 2 0 (G) that are twice continuously differentiable and compactly supported in G. Any admissible function may be written u = vw using the harmonic function (1) and v ∈ C 2 0 (G). The divergence theorem shows G |du| 2 da = G w 2 |dv| 2 da.
Making the change of variable x = 1 2 (1 − cos r) transforms the ODE to the hypergeometric equation on [0, 1] The solution to the hypergeometric equation is Gauß's ordinary hypergeometric function, given by We find the eigenvalue by a shooting method. Given r * , λ * is the first positive root of the function Consider the example of the geodesic triangle T ⊂ S 2 which is a face of the regular tetrahedral tessellation, whose vertices in the unit sphere could be taken as 1 √ 3 , ± 2 3 , 0 and − 1 √ 3 , 0, ± 2 3 . The distance between vertices is ε = cos −1 − 1 3 . The diameter, which equals the distance from vertex to center of the opposite edge, is δ = cos −1 − 1 √ 3 . T fits inside a wedge sharing a vertex of angle 2π 3 . Writing we find r(θ) = π 2 + Atn cos(θ − π 3 ) √ 2 .
λ 1 (T ) was found numerically in [RT]. Using the computer algebra system Maple c , we numerically integrate I(T ) = π/α 0 Z(r(θ)) sin 2 (αθ) dθ and solve π 2α Z(r * ) = I(T ) for r * and (23) for λ * to get the other values in the T line in Table 1. To avoid the quadrature, we observe the estimate Proof. Details are provided in [RT]. Finiteness of the expectation of τ 4 follows if it can be shown that λ 1 (T ) > 5.101267527. The lower eigenvalue bound is given by Theorem 1 applied to T depends on either the numerical integration of I(T ) or its upper bound by the quadrature free estimate of (24).