Enea Parini: Optimal shape for an isoperimetric quantity related to Cheeger’s inequality

We consider the minimization of the functional \[ J(\Omega) := \frac{\lambda_1(\Omega)}{h_1(\Omega)^2},\] where $\lambda_1(\Omega)$ is the first eigenvalue of the Laplacian under Dirichlet boundary conditions, and $h_1(\Omega)$ is the Cheeger constant of $\Omega$, in the class $\mathcal{K}$ of planar convex sets. We provide a lower bound which improves the generic bound given by Cheeger's inequality, we show the existence of a minimizer, and give some optimality conditions, which can be summarized by saying that a minimizer must have a ``polygonal shape''. A crucial role is played by a new result on partially overdetermined boundary value problems, which relies on a recent paper by Fragal\`{a} and Gazzola. We also give the sharp upper bound for $J$ in $\mathcal{K}$, we show that the supremum is not attained, and we provide explicit maximizing sequences.