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 $Ω\subset\mathbb R^d$, with boundary data given by the indicator of a closed set. At a smooth boundary point where the data vanish, we study the upper and lower inward normal Dini derivatives as $p\downarrow1$. We