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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)

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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).

Item Type: Article
Additional Information: 47 Pages
Uncontrolled Keywords: Gap series; p-harmonic measure; p-harmonic function; radial limits; Fatou theorem
Divisions: Faculty of Science and Health
Faculty of Science and Health > Mathematical Sciences, Department of
SWORD Depositor: Elements
Depositing User: Elements
Date Deposited: 16 Sep 2021 10:10
Last Modified: 06 Jun 2022 14:48
URI: http://repository.essex.ac.uk/id/eprint/31102

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