Convex planar isoperimetric index-gap conjecture
Establish the existence of a universal constant C ∈ (0,1) such that for every bounded convex planar domain Ω with perimeter P and area A, the Laplacian Dirichlet and Neumann spectra satisfy λ_k(Ω) ≥ μ_{k + ⌊ C P √k /(π A) ⌋}(Ω) for all positive integers k.
References
Inspired by the result for rectangles in Theorem 3.1 already mentioned above, and by the results in Section 3.3 for general planar domains, it is also possible to formulate other conjectures with a sequence p depending on the isoperimetric constant of the given domain. Again, this cannot hold either in general or even for convex domains, as both the examples given in [hatc] for nonconvex domains, and those for the disk in Section 4 below show. We thus restrict the conjecture to convex domains, and, for simplicity, state it only in the planar case. There exists a constant C in (0,1) such that the Dirichlet and Neumann eigenvalues of a bounded convex planar domain Ω with perimeter P and area A satisfy the inequality λ{k}(Ω) ≥ μ{k + ⌊ C P √k /(π A) ⌋}(Ω) for all k∈N.