Lower bounds for the first eigenvalue of the laplacian, with dirichlet boundary conditions and a theorem of hayman

Abstract

Lower bounds for the first eigenvalue of the eigenvalue problem Δu + λu = 0 in ω, u = 0 on ω, are given. Specifically, it is proved that if for all × ε ω, the ball of radius ρ centered at x intersects the complement of ω in a set of capacity greater than or equal to δ, then the first eigenvalue λ₁ satisifes the inequality