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

Let Ω be a Lipschitz bounded domain of R N , N ≥ 2 , and let u p ∈ W 0 1 , p ( Ω ) denote the p -torsion function of Ω , p > 1. It is observed that the value 1 for the Cheeger constant h ( Ω ) is threshold with respect to the asymptotic behavior of u p , as p → 1 + , in the following sense: when h ( Ω ) > 1 , one has lim p → 1 + u p L ∞ ( Ω ) = 0 , and when h ( Ω ) < 1 , one has lim p → 1 + u p L ∞ ( Ω ) = ∞ . In the case h ( Ω ) = 1 , it is proved that lim sup p → 1 + u p L ∞ ( Ω ) < ∞ . For a radial annulus Ω a , b , with inner radius a and outer radius b , it is proved that lim p → 1 + u p L ∞ ( Ω a , b ) = 0 when h ( Ω a , b ) = 1 .