A monotonicity property of the p-torsional rigidity

Abstract For a bounded domain Ω ⊂ R N , N ≥ 2 , and a real number p > 1 , we denote by u p the p -torsion function on Ω , that is the solution of the torsional creep problem Δ p u = − 1 in Ω , u = 0 on ∂ Ω , where Δ p u ≔ d i v ( ∇ u p − 2 ∇ u ) is the p -Laplace operator. Our aim is to investigate some monotonicity properties for the p -torsional rigidity on Ω , defined as T p Ω ≔ ∫ Ω u p d x . More precisely, we first show that there exists T ∈ 0 , 1 such that for each open, bounded, convex domain Ω ⊂ R N , with smooth boundary and δ Ω ≤ T , where δ Ω represents the average integral on Ω of the distance function to the boundary of Ω , the function p → T p ; Ω ≔ Ω p − 1 T p Ω 1 − p is increasing on 1 , ∞ . Moreover, we also show that for any real number s > T , there exists an open, bounded, convex domain Ω ⊂ R N , with smooth boundary and δ Ω = s , such that the function p → T p ; Ω is not a monotone function of p ∈ ( 1 , ∞ ) . Finally, we use this result to get a new variational characterization of T ( p ; Ω ) , in the case when δ Ω is small enough.