On the monotonicity of the principal frequency of the p-Laplacian

Abstract

Abstract For any fixed integer D > 1 {D>1} we show that there exists M ∈ [ e - 1 , 1 ] {M\in[e^{-1},1]} such that for any open, bounded, convex domain Ω ⊂ ℝ D {\Omega\subset{\mathbb{R}}^{D}} with smooth boundary for which the maximum of the distance function to the boundary of Ω is less than or equal to M, the principal frequency of the p-Laplacian on Ω is an increasing function of p on ( 1 , ∞ ) {(1,\infty)} . Moreover, for any real number s > M {s>M} there exists an open, bounded, convex domain Ω ⊂ ℝ D {\Omega\subset{\mathbb{R}}^{D}} with smooth boundary which has the maximum of the distance function to the boundary of Ω equal to s such that the principal frequency of the p-Laplacian is not a monotone function of p ∈ ( 1 , ∞ ) {p\in(1,\infty)} .