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

Akman, Murat and Lewis, John and Vogel, Andrew (2022) 'Failure of Fatou type theorems for solutions to PDE of p-Laplace type in domains with flat boundaries.' Communications in Partial Differential Equations. pp. 1-47. ISSN 0360-5302 (In Press)

 Preview
Text
ALV_Wolff_Final.pdf - Submitted Version

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).