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 for 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, pp. 85–133], [LT, pp. 37–40], [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, recall that the round metric is given by ds = dρ + sinρ dθ. Let W = {(ρ, θ) : 0 ≤ θ ≤ π/α, 0 ≤ ρ < π} be the sector in S 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) := {(ρ, θ) : 0 ≤ θ ≤ π/α, 0 ≤ ρ ≤ r}. Observe that w = tan (ρ 2 ) sinαθ (1) 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 λ1(G) ≥ λ1(S(r)), (2) ∗AMS Subject classification. Primary: 35P15. 1 where r∗ is chosen such that I(G) = ∫