The eigenvalue problem for the Laplacian equations

Abstract

This article studies the Dirichlet eigenvalue problem for the Laplacian equations Δu =-λ u, χ∈ Ω, u=0, δ∈Ω, where Ω ⊂ R n is a smooth bounded convex domain. By using the method of appropriate barrier function combined with the maximum principle, authors obtain a sharp lower bound of the difference of the first two eigenvalues for the Dirichlet eigenvalue problem. This study improves the result of S.T. Yau et al.