Mihai Mihăilescu
University of Craiova, Craiova, Romania & Gheorghe Mihoc-Caius Iacob Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy, Bucharest, Romania

For any bounded and convex set $\Omega \subset \mathbb{R}^{N}$ ($N\geq 2$) with smooth boundary, $\partial\Omega$, and any real number $p>1,$ we denote by $u_{p}$ the $p$-torsion function on $\Omega $, that is the solution of the torsional creep problem $\Delta_{p}u=-1$ in $\Omega $, $u=0$ on $\partial \Omega $, where $\Delta _{p}u:=div( \left\vert \nabla u\right\vert ^{p-2}\nabla u) $ is the $p$-Laplace operator.

Our aim is to investigate the monotonicity with respect to $p$ for the $p$-torsional rigidity on $\Omega $, defined as $T_{p}\left( \Omega \right) :=\int_{\Omega }u_{p}dx$. More precisely, we establish that there exist two constants $D_1\in\left[\frac{1}{2},e^{\frac{-1}{N+1}}\right]$ and $D_2\in\left[1,N\right]$ such that for each bounded and convex set $\Omega\subset \mathbb{R}^{N}$ with $\frac{|\partial\Omega|}{|\Omega|}\leq D_1$ the function $p\mapsto T_p(\Omega)$ is decreasing on $\left( 1,\infty \right) $ while for each bounded and convex set $\Omega\subset \mathbb{R}^{N}$ with $\frac{|\partial\Omega|}{|\Omega|}\geq D_2$ the function $p\mapsto T_p(\Omega)$ is increasing on $\left( 1,\infty \right) $. Moreover, for each real number $s\in(D_1,D_2)$ there exists a bounded and convex set $\Omega\subset\mathbb{R}^{N}$ with $\frac{|\partial\Omega|}{|\Omega|}=s$ such that the function $p\mapsto T_p(\Omega)$ is not monotone on $(1,\infty)$. This is a joint work with Cristian Enache and Denisa Stancu-Dumitru.