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

Abstract

Let ${\mathbb{R}}^n$ denote Euclidean $n$ space and given $k$ a positive integer let ${\Lambda }_k\subset {\mathbb{R}}^n$, $1\le k<n-1,n\ge 3,$ 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).

47 Pages