Sharp ratios for low-index Neumann eigenvalues on convex domains
Abstract
Let Ω⊂RN be a bounded open convex set, and let 0=μ0(Ω)<μ1(Ω) μ2(Ω)·s be the Neumann eigenvalues of the Laplacian, repeated according to multiplicity. We prove the sharp bounds μ2(Ω) 4μ1(Ω), μ3(Ω) 9μ1(Ω). 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.
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