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

Abstract

For each open, bounded and convex domain $\Omega \subset {\mathbb{R}}^D,$ $D\ge 2$, 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 ${\Delta }_{p}u:=\mathrm{div}({\left|\nabla u\right|}^{p-2}\nabla u)$ 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 $T\left(p;\Omega \right):={\left|\Omega \right|}^{p-1}{T}_p{\left(\Omega \right)}^{1-p}$, 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 $T(p;{\Omega }_b)\le T(p;{\Omega }_a)$, for all $ Ω_a\in\PP^D(a)$, and $Ω_b\in\PP^D(b)$, if and only if ${\gamma }_{D,p}b\ge a$, 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 $p\to 1^{+}$ and $p\to \infty$, 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.