Comparison results for the $ p $-torsional rigidity on convex domains

For each open, bounded and convex domain $ \Omega \subset \mathbb{R}^{D}, $ $ D\geq 2 $, and each real number $ p > 1, $ we denote by $ u_{p} $ the $ p $-torsion function on $ \Omega $, i.e., the solution of the torsional creep problem $ \Delta_{p}u = -1 $ in $ \Omega $, $ u = 0 $ on $ \partial \Omega $, where $ \Delta _{p}u: = \operatorname{div}(\left\vert \nabla u\right\vert ^{p-2}\nabla u) $ is the $ p $-Laplacian. Let $ T_p(\Omega) $ be the $ p $-torsional rigidity on $ \Omega $, defined as $ T_{p}\left(\Omega \right) : = \int_{\Omega }u_{p}dx $. Define $ T\left(p; \Omega \right) : = \left\vert \Omega \right\vert ^{p-1}T_{p}\left(\Omega \right) ^{1-p} $, where $ |\Omega| $ stands for the Lebesgue measure of $ \Omega $. The main purpose of this paper is to compare the values of $ T(p; \Omega) $ for bounded convex domains having different inradii. We prove that for any $ 0 < a < b $ there exists an explicit constant $ \gamma_{D, p}\in[1/D, 1) $, depending only on the dimension $ D $ and the parameter $ p $, such that $ T(p; \Omega_b)\leq T(p; \Omega_a) $, for all $ \Omega_a\in\mathbb{P}^D(a) $, and $ \Omega_b\in\mathbb{P}^D(b) $, if and only if $ \gamma_{D, p}b\geq a $, where $ \mathbb{P}^D(r) $ denotes the family of convex bounded domains in $ \mathbb{R}^D $ of inradius $ r $. In addition, we discuss the asymptotic equality case, the limiting regimes $ p\rightarrow 1^+ $ and $ p\rightarrow \infty $, and the behaviour of the scale-invariant functional on model families such as rectangles, orthotopes, ellipses, and triangles. We also derive a fixed-volume monotonicity consequence for the unnormalised torsional rigidity, parameterized by the inradius.