More Bounds on Eigenvalue Ratios for Dirichlet Laplacians in N Dimensions
Mark S. Ashbaugh, Rafael D. Benguria · SIAM Journal on Mathematical Analysis · 1993
The authors investigate bounds for various combinations of the low eigenvalues of the Laplacian with Dirichlet boundary conditions on a bounded domain $\Omega \subset \mathbb{R}^n $. These investigations continue and expand upon earlier work of Payne, Pólya, Weinberger, Brands, Chiti, and the authors of this present paper. In particular, the authors generalize and extend to the n-dimensional setting various bounds of Payne, Pólya, Weinberger, Brands, and Chiti and examine their consequences and interrelationships in detail. This includes comparing the asymptotic forms of the various bounds as the dimension n becomes large. The authors also present various extensions and consequences of their recent proof of the Payne–Pólya–Weinberger conjecture, including the proof of a second conjecture of Payne, Pólya, and Weinberger under an added symmetry condition.