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

TL;DR

This paper investigates how boundary derivatives of p-harmonic functions behave as p approaches 1, revealing nonlocal influences on explosion or decay rates with specific examples and conditions.

Contribution

It provides new conditions for boundary derivative explosion or decay rates as p tends to 1, highlighting nonlocal effects and presenting a critical example.

Findings

Boundary derivatives can explode at rate C_Ω/(p-1) or decay at rate exp(-c_Ω/(p-1))

Explosion or decay depends on nonlocal properties, not just local data

A cylinder example shows explosion at rate C_d/√(p-1)

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 $\frac{C_d}{\sqrt{p-1}}$.