INEQUALITIES FOR EIGENVALUES OF THE LAPLACIAN
Qing-Ming Cheng, Xuerong Qi · arXiv (Cornell University) · 2011
For a bounded domain with a piecewise smooth boundary in an n-dimensional Euclidean space R n , we study eigenvalues of the Dirichlet eigenvalue problem of the Laplacian. By making use of a fact that eigenfunctions form an orthonormal basis of L 2 () in place of the Rayleigh-Ritz formula, we obtain inequalities for eigenvalues of the Laplacian. In particular, we study a conjecture of Payne, Polya and Weinberger on lower order eigenvalues. Our results will become an important progress for solving this conjecture.