Sharp ratios for low-index Neumann eigenvalues on convex domains

Abstract

Let $\hat{I}\text{c◯}\subset {\mathbb{R}}^N$ be a bounded open convex set, and let $0=\hat{I}{¼}_0(\hat{I}\text{c◯})<\hat{I}{¼}_1(\hat{I}\text{c◯})\le \hat{I}{¼}_2(\hat{I}\text{c◯})\le \cdots$ be the Neumann eigenvalues of the Laplacian, repeated according to multiplicity. We prove the sharp bounds $$\hat{I}{¼}_2(\hat{I}\text{c◯})\le 4\hat{I}{¼}_1(\hat{I}\text{c◯}),\hat{I}{¼}_3(\hat{I}\text{c◯})\le 9\hat{I}{¼}_1(\hat{I}\text{c◯}).$$ The first estimate resolves a problem attributed to Henrot, while the second gives the next sharp case predicted by the one-dimensional model. The constants are optimal in every dimension.

15 pages. Comments and suggestions are welcome