Explosion versus decay for boundary derivatives of $p$-harmonic functions as $p$ tends to 1: nonlocality

We consider the Dirichlet problem for the $p$-Laplacian on a bounded Lipschitz domain $\mathrm{\Omega }\subset {\mathbb{R}}^d$ with a $\{0,1\}$-valued function as the boundary condition and study the dependence of the boundary derivative on $p$ as $p\downarrow 1$. We provide sufficient conditions for the derivative to explode at rate $\frac{C_{\mathrm{\Omega }}}{p-1}$ and to decay at rate $\exp (-\frac{c_{\mathrm{\Omega }}}{p-1})$. Surprisingly, whether explosion or decay occurs is not determined locally. We also present a critical example of a cylinder where this derivative explodes at rate $\frac{C_d}{\sqrt{p-1}}$.