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

Abstract

We consider the Dirichlet problem for the $p$-Laplacian on a bounded Lipschitz domain $\Omega \subset \mathbb{R}^d$ with a $\{0,1\}$-valued function as the boundary condition and study the dependence of the boundary derivative on $p$ as $p\downarrow1$. We provide sufficient conditions for the derivative to explode at rate $\frac{C_\Omega}{p-1}$ and to decay at rate $\exp(-\frac{c_\Omega}{p-1})$. 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 $\frac{C_d}{\sqrt{p-1}}$.