On the p-torsion functions of an annulus

Let p>1 and denote, respectively, by u_p and h(Ω_a,b), the p-torsion function and the Cheeger constant of the annulus $\Omega_{a,b}=\{x\in\mathbb{R}^{N}\dvt a<\vert x\vert <b\}$ , N>1. Thus, u_p is the solution of the p-torsional creep problem \[\cases{-\operatorname{div}\bigl(\vert \nabla u\vert ^{p-2}\nablau\bigr)=1&in $\Omega_{a,b}$,\cru=0&on $\partial\Omega_{a,b}$}\] and \[h(\Omega_{a,b}):=\min \biggl\{\frac{\vert \partial E\vert }{\vert E\vert }\dvt E\subset\overline{\Omega_{a,b}}\biggr\},\] where |∂E| and |E| denote, respectively, the (N−1)-dimensional Lebesgue perimeter of ∂E in $\mathbb{R}^{N}$ and the N-dimensional Lebesgue volume of the smooth subset $E\subset\overline{\Omega_{a,b}}$ .

We prove that \[\lim_{p\rightarrow1^{+}}\Vert u_{p}\Vert _{\infty}^{1-p}=\lim_{p\rightarrow1^{+}}\Vert \nabla u_{p}\Vert _{\infty}^{1-p}=N\frac{b^{N-1}+a^{N-1}}{b^{N}-a^{N}}=\frac{\vert \partial\Omega_{a,b}\vert }{\vert \Omega_{a,b}\vert }\] and combine this fact with a characterization of the Cheeger constant that we proved in a previous paper, to give a new proof of the calibrability of Ω_a,b, that is, $h(\Omega_{a,b})=\frac{\vert \partial\Omega_{a,b}\vert }{\vert \Omega_{a,b}\vert }$ .

Moreover, we prove that u_p is concave and satisfies lim _p→1⁺(u_p(x)/‖u_p‖_∞)=1, uniformly in the set a+ε≤|x|≤b−ε, for all ε>0 sufficiently small.

Our results rely on estimates for m_p, the radius of the sphere on which u_p assumes its maximum value. We derive these estimates by combining Pohozaev's identity for the p-torsional creep problem with a kind of l'Hôpital rule for monotonicity.