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).
Categories: math.AP