A monotonicity property of the p-torsional rigidity. (July 2021)

• Record Type: Journal Article
• Authors: Enache, Cristian; Mihăilescu, Mihai
• Is Part Of: Nonlinear analysis. Volume 208(2021)
• Publication Date: 2021-07
• Subjects: 35Q74 -- 47J05 -- 47J20 -- 49J40 -- 49S05; p-Laplacian -- p-torsional rigidity -- Distance function to the boundary; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2021.112326
• Languages: English
• ISSNs: 0362-546X

Abstract

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.