Failure of Fatou type theorems for solutions to PDE of $p$-Laplace type in domains with flat boundaries

TL;DR

This paper investigates the boundary behavior of p-harmonic functions near flat boundaries in higher dimensions, showing the failure of Fatou type theorems in these settings and extending previous two-dimensional results.

Contribution

It extends Wolff's work on Fatou theorem failures from 2D to higher dimensions with flat boundaries for p-harmonic functions.

Findings

Failure of Fatou theorems in higher dimensions with flat boundaries.

Characterization of the Martin boundary for p-harmonic functions in these domains.

Extension of results to -harmonic functions.

Abstract

Let $ \mathbb{R}^{n} $ denote Euclidean $ n $ space and given $k$ a positive integer let $ \Lambda_k \subset \mathbb{R}^{n} $, $ 1 \leq k < n - 1, n \geq 3, $ be a $k$-dimensional plane with $ 0 \in \Lambda_k.$ If $n-k < p <\infty$, we first study the Martin boundary problem for solutions to the $p$-Laplace equation (called $p$-harmonic functions) in $ \mathbb{R}^{n} \setminus \Lambda_k $ relative to $ \{0\}. $ We then use the results from our study to extend the work of Wolff on the failure of Fatou type theorems for $p$-harmonic functions in $ \mathbb{R}^{2}_+ $ to $p$-harmonic functions in $ \mathbb{R}^{n} \setminus \Lambda_k $ when $ n-k < p <\infty$. Finally, we discuss generalizations of our work to solutions of $ p $-Laplace type PDE (called $ \mathcal{A}$-harmonic functions).