We discuss the minimization of a Kohler-Jobin type scale-invariant functional among open, convex, bounded sets, namely {\$}{\$} {\backslash}min {\backslash}Big {\backslash}{\{} T{\_}2({\backslash}Omega ) ^{\{}{\backslash}frac{\{}1{\}}{\{}N+2{\}}{\}}h{\_}1({\backslash}Omega ) : {\backslash}Omega {\backslash}subset {\backslash}mathbb R^N,{\backslash}text {\{} open, convex, bounded{\}}{\backslash}Big {\backslash}{\}}{\backslash}, {\$}{\$}min{\{}T2($\Omega$)1N+2h1($\Omega$):$\Omega$⊂RN,open, convex, bounded{\}}where {\$}{\$}T{\_}2({\backslash}Omega ){\$}{\$}T2($\Omega$)denotes the torsional rigidity of a set {\$}{\$}{\backslash}Omega {\$}{\$}$\Omega$and {\$}{\$}h{\_}1({\backslash}Omega ){\$}{\$}h1($\Omega$)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.
On a Cheeger--Kohler-Jobin Inequality
Ilaria Lucardesi;
2024-01-01
Abstract
We discuss the minimization of a Kohler-Jobin type scale-invariant functional among open, convex, bounded sets, namely {\$}{\$} {\backslash}min {\backslash}Big {\backslash}{\{} T{\_}2({\backslash}Omega ) ^{\{}{\backslash}frac{\{}1{\}}{\{}N+2{\}}{\}}h{\_}1({\backslash}Omega ) : {\backslash}Omega {\backslash}subset {\backslash}mathbb R^N,{\backslash}text {\{} open, convex, bounded{\}}{\backslash}Big {\backslash}{\}}{\backslash}, {\$}{\$}min{\{}T2($\Omega$)1N+2h1($\Omega$):$\Omega$⊂RN,open, convex, bounded{\}}where {\$}{\$}T{\_}2({\backslash}Omega ){\$}{\$}T2($\Omega$)denotes the torsional rigidity of a set {\$}{\$}{\backslash}Omega {\$}{\$}$\Omega$and {\$}{\$}h{\_}1({\backslash}Omega ){\$}{\$}h1($\Omega$)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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
