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

Abstract

Let Ω be a Lipschitz bounded domain of RN, N ≥ 2, and let up ∈ W1,p 0 (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 up, as p → 1+, in the following sense: when h(Ω) > 1, one has limp→1+ up L∞(Ω) = 0, and when h(Ω) < 1, one has limp→1+ up L∞(Ω) = ∞. In the case h(Ω) = 1, it is proved that lim supp→1+ up L∞(Ω) < ∞. For a radial annulus Ωa,b, with inner radius a and outer radius b, it is proved that limp→1+ up L∞(Ωa,b) = 0 when h(Ωa,b) = 1.