On a Cheeger--Kohler-Jobin inequality

Ilaria Lucardesi; Dario Mazzoleni; Berardo Ruffini

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arXiv · 1806.01549

On a Cheeger--Kohler-Jobin inequality

Ilaria Lucardesi·Dario Mazzoleni·Berardo Ruffini

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Abstract

We discuss the minimization of a Kohler-Jobin type scale-invariant functional among open, convex, bounded sets, namely $\min T_2(\Omega) ^{\frac{1}{N+2}}h_1(\Omega)$ among open convex bounded sets $\Omega \subset \mathbb R^N$, 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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Ilaria Lucardesi, Dario Mazzoleni, Berardo Ruffini. 2018-06-05. On a Cheeger--Kohler-Jobin inequality. https://arxiv.org/abs/1806.01549

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