For a bounded domain $\Omega \subset {\mathbb{R}}^N,$ $N\ge 2$, and a 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 ${\Delta }_{p}u≔div({\left|\nabla u\right|}^{p-2}\nabla u)$ is the $p$-Laplace operator. Our aim is to investigate some monotonicity properties for the $p$-torsional rigidity on $\Omega$, defined as $T_p\left(\Omega \right)≔{\int }_{\Omega }{u}_{p}dx$. More precisely, we first show that there exists $T\in \left(0,1\right]$ such that for each open, bounded, convex domain $\Omega \subset {\mathbb{R}}^N$, with smooth boundary and $\delta \left(\Omega \right)\le T$, where $\delta \left(\Omega \right)$ represents the average integral on $\Omega$ of the distance function to the boundary of $\Omega$, the function $p\to T\left(p;\Omega \right)≔{\left|\Omega \right|}^{p-1}{T}_p{\left(\Omega \right)}^{1-p}$ is increasing on $\left(1,\infty \right)$. Moreover, we also show that for any real number $s>T,$ there exists an open, bounded, convex domain $\Omega \subset {\mathbb{R}}^N,$ with smooth boundary and $\delta \left(\Omega \right)=s$, such that the function $p\to T\left(p;\Omega \right)$ is not a monotone function of $p\in (1,\infty )$. Finally, we use this result to get a new variational characterization of $T(p;\Omega )$, in the case when $\delta \left(\Omega \right)$ is small enough.

对于有界域 $\Omega \subset {\mathbb{[R}}^{ñ}，$ $ñ\ge \mathrm{2个}$和一个实数 $p>\mathrm{1个}，$ 我们用 ${ü}_p$ 这 $p$扭转功能开启$\Omega$，这是扭转蠕变问题的解决方案 ${\Delta }_pü=-\mathrm{1个}$ 在 $\Omega$， $ü=0$ 在 $\partial \Omega$， 在哪里 ${\Delta }_pü≔d\mathrm{一世}v（{\left|\nabla ü\right|}^{p-\mathrm{2个}}\nabla ü）$ 是个 $p$-拉普拉斯运算符。我们的目的是研究$p$-torsional刚性上$\Omega$， 定义为 ${Ť}_p\left(\Omega \right)≔{\int }_{\Omega }{ü}_{p}dX$。更确切地说，我们首先证明存在$Ť\in \left(0，\mathrm{1个}\right]$ 这样对于每个开放的，有界的，凸的域 $\Omega \subset {\mathbb{[R}}^{ñ}$，具有平滑的边界和 $\delta \left(\Omega \right)\le Ť$， 在哪里 $\delta \left(\Omega \right)$ 代表上的平均积分 $\Omega$ 距离函数到边界的距离 $\Omega$， 功能 $p\to Ť\left(p;\Omega \right)≔{\left|\Omega \right|}^{p-\mathrm{1个}}{Ť}_p{\left(\Omega \right)}^{\mathrm{1个}-p}$ 在增加 $\left(\mathrm{1个}，\infty \right)$。此外，我们还表明对于任何实数$s>Ť，$ 存在一个开放的，有界的，凸的域 $\Omega \subset {\mathbb{[R}}^{ñ}，$ 具有光滑的边界和 $\delta \left(\Omega \right)=s$，这样的功能 $p\to Ť\left(p;\Omega \right)$ 不是...的单调函数 $p\in （\mathrm{1个}，\infty ）$。最后，我们使用这个结果来获得$Ť（p;\Omega ）$，如果 $\delta \left(\Omega \right)$ 足够小。