Preprint
Inserted: 6 mar 2023
Last Updated: 6 mar 2023
Year: 2023
Abstract:
We discuss the minimization of a Kohler-Jobin type scale-invariant functional among open, convex, bounded sets, namely \[ \min\Big\{ T_2(\Omega) ^{\frac{1}{N+2}}h_1(\Omega) : \Omega\subset\mathbb{R}^N,\text{ open, convex, bounded}\Big\}\, \] where $T_2(\Omega)$ denotes the torsional rigidity of a set $\Omega$ and $h_1(\Omega)$ its Cheeger constant. We prove the existence of an optimal set and we conjecture that the ball is the unique minimizer. We provide a sufficient condition for the validity of the conjecture, and an application of the conjecture to prove a quantitative inequality for the Cheeger constant. We also show lack of existence for the problem above among several other classes of sets. As a side result we discuss the equivalence of the several definitions of Cheeger constants present in the literature and show a quite general class of sets for which those are equivalent.
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