Rate of convergence of the $p$-torsion function to the distance function

We prove that the Dirichlet $p$-torsion function $u_p$ converges to the distance to the boundary $d$ at the rate $O(1\mathrm{/}p)$ in the uniform norm, on every bounded domain and with a constant depending only on the dimension and the diameter. The upper bound comes from a radial barrier with its pole at a boundary point, which is an admissible comparison function for $p>n$ and needs no boundary regularity; the lower bound comes from an inscribed ball. On the ball the error equals $(1+\log n)\mathrm{/}p+O(p^{-2})$, so the order $1\mathrm{/}p$ cannot be improved in general. Under a uniform exterior ball condition an annular barrier gives $u_p\le K_{\mathrm{\Omega }}^{1\mathrm{/}(p-1)}d$ for every $p>1$, with $K_{\mathrm{\Omega }}$ explicit in the dimension, the diameter and the exterior radius; on convex sets the constant is the diameter, and a multiple of $d$ is a supersolution whenever $-\mathrm{\Delta }d$ has a positive distributional lower bound. On $C^2$ domains, the same barrier and the classical gradient maximum principle give $\text{norm}{\mathrm{\nabla }{u}_p}_{L^{\mathrm{\infty }}}^{p-1}\le K_{\mathrm{\Omega }}$. When $-\mathrm{\Delta }d$ is a measure of finite total variation, which we prove for $C^1$ domains with a uniform exterior ball, the gradients converge in $L^q$ at the rate $O(p^{-1\mathrm{/}2})$ for $q\le 2$ and $O(p^{-1\mathrm{/}q})$ for $q\ge 2$; these exponents are not claimed to be sharp. The explicit ball profile shows that the gradients do not converge uniformly.