A monotonicity property of the p-torsional rigidity. (July 2021) - Enache, Cristian; Mihăilescu, Mihai
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.