LAUSR

We deal with monotonicity with respect to \begin{document}$ p $\end{document} of the first positive eigenvalue of the \begin{document}$ p $\end{document} -Laplace operator on \begin{document}$ \Omega $\end{document} subject to the homogeneous Neumann boundary condition. For any fixed integer \begin{document}$ D>1 $\end{document} we show that there exists \begin{document}$ M\in[2 e^{-1}, 2] $\end{document} such that for any open, bounded, convex domain \begin{document}$ \Omega\subset{{\mathbb R}}^D $\end{document} with smooth boundary for which the diameter of \begin{document}$ \Omega $\end{document} is less than or equal to \begin{document}$ M $\end{document} , the first positive eigenvalue of the \begin{document}$ p $\end{document} -Laplace operator on \begin{document}$ \Omega $\end{document} subject to the homogeneous Neumann boundary condition is an increasing function of \begin{document}$ p $\end{document} on \begin{document}$ (1, \infty) $\end{document} . Moreover, for each real number \begin{document}$ s>M $\end{document} there exists a sequence of open, bounded, convex domains \begin{document}$ \{\Omega_n\}_n\subset{{\mathbb R}}^D $\end{document} with smooth boundaries for which the sequence of the diameters of \begin{document}$ \Omega_n $\end{document} converges to \begin{document}$ s $\end{document} , as \begin{document}$ n\rightarrow\infty $\end{document} , and for each \begin{document}$ n $\end{document} large enough the first positive eigenvalue of the \begin{document}$ p $\end{document} -Laplace operator on \begin{document}$ \Omega_n $\end{document} subject to the homogeneous Neumann boundary condition is not a monotone function of \begin{document}$ p $\end{document} on \begin{document}$ (1, \infty) $\end{document} .