Reverse Cheeger inequality for planar convex sets

We prove the sharp inequality $J(Ω) := \frac{λ_1(Ω)}{h_1(Ω)^2} < \frac{π^2}{4},$ where $Ω$ is any planar, convex set, $λ_1(Ω)$ is the first eigenvalue of the Laplacian under Dirichlet boundary conditions, and $h_1(Ω)$ is the Cheeger constant of $Ω$. The value on the right-hand side is optimal, and any sequence of convex sets with fixed volume and diameter tending to infinity is a maximizing sequence. Morever, we discuss the minimization of $J$ in the same class of subsets: we provide a lower bound which improves the generic bound given by Cheeger's inequality, we show the existence of a minimizer, and we give some optimality conditions.

No reviews yet.

No comments yet.