Affiliations: Departamento de Matemática, Universidade Federal de Minas Gerais, Belo Horizonte, Minas Gerais, 30.123.970, Brazil. E-mails: [email protected], [email protected]
Abstract: Let p>1 and denote, respectively, by up and h(Ωa,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, up 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 Ωa,b, that is, $h(\Omega_{a,b})=\frac{\vert \partial\Omega_{a,b}\vert }{\vert \Omega_{a,b}\vert }$. Moreover, we prove that up is concave and satisfies lim p→1+(up(x)/‖up‖∞)=1, uniformly in the set a+ε≤|x|≤b−ε, for all ε>0 sufficiently small. Our results rely on estimates for mp, the radius of the sphere on which up 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.
Keywords: annulus, Cheeger constant, p-Laplacian, p-torsion functions, torsional creep problem
