Asymptotic behavior of the p-torsion functions as p goes to 1

Let ({Omega}) be a Lipschitz bounded domain of ({mathbb{R}^N}), ({Ngeq2}), and let ({u_pin W_0^{1,p}(Omega)}) denote the p-torsion function of ({Omega}), p > 1. It is observed that the value 1 for the Cheeger constant ({h(Omega)}) is threshold with respect to the asymptotic behavior of u_p, as ({prightarrow 1^+}), in the following sense: when ({h(Omega) > 1}), one has ({lim_{prightarrow 1^+}left|u_{p}right| _{L^infty(Omega)}=0}), and when ({h(Omega) < 1}), one has ({lim_{prightarrow 1^+}left|u_pright| _{L^infty(Omega)}=infty}). In the case ({h(Omega)=1}), it is proved that ({limsup_{prightarrow1^+}left|u_pright|_{L^infty(Omega)}a and outer radius b, it is proved that ({lim_{prightarrow 1^+}left|u_pright| _{L^infty(Omega_{a,b})}=0}) when ({h(Omega_{a,b})=1}).