Abstract:

This article studies the Dirichlet eigenvalue problem for the Laplacian equations △u=-λu, $x\in \Omega$, $u=0$, $x\in\partial\Omega$, where $\Omega\subset R^{n}$ 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.

Key words: Dirichlet eigenvalue problem, gradient estimate, maximum principle, barrier function