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

Abstract

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 $p$\emph{-torsion function} on $\Omega $, i.e. the solution of the \emph{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 $p$\emph{-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 a 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\PP^D(a)$, and $\Omega_b\in\PP^D(b)$, if and only if $\gamma_{D,p}b\geq a$, where $\PP^D(r)$ 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 sharpness of our bounds on model families such as rectangles, orthotopes, ellipses, and triangles}. We also derive a Saint-Venant type comparison result under additional geometric constraints, as a direct consequence of our main theorem.