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 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 $p\downarrow 1$. We give sufficient geometric conditions for these derivatives to be of order $(p-1{)}^{-1}$, and other conditions under which they decay exponentially in $1/(p-1)$. Whether explosion or decay occurs is not determined locally: on one fixed planar domain with real-analytic boundary, changing the data on a set of arbitrarily small total length, uniformly separated from the observation point, changes explosion into exponential decay. We also exhibit a critical example of a cylinder in ${\mathbb{R}}^{d+1}$ where the inward normal derivative is of order $\sqrt{d/(p-1)}$.

arXiv comment15pages, 9 figures