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Let $ \mathbb{R}^{n} $ denote Euclidean $ n $ space and given $k$ a positiveinteger 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 $n-k < p <\infty$, wefirst study the Martin boundary problem for solutions to the $p$-Laplaceequation (called $p$-harmonic functions) in $ \mathbb{R}^{n} \setminus\Lambda_k $ relative to $ \{0\}. $ We then use the results from our study toextend the work of Wolff on the failure of Fatou type theorems for $p$-harmonicfunctions in $ \mathbb{R}^{2}_+ $ to $p$-harmonic functions in $ \mathbb{R}^{n}\setminus \Lambda_k $ when $ n-k < p <\infty$. Finally, we discussgeneralizations of our work to solutions of $ p $-Laplace type PDE (called $\mathcal{A}$-harmonic functions).

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