Failure of Fatou type theorems for solutions to PDE of p-Laplace type in domains with flat boundaries. Issue 7 (3rd July 2022) - Akman, Murat
Abstract: Let R n denote Euclidean n space and given k a positive integer let Λ k ⊂ R n, 1 ≤ k < n − 1, n ≥ 3, be a k -dimensional plane with 0 ∈ Λ k . If n − k < p < ∞, we first study the Martin boundary problem for solutions to the p-Laplace equation (called p-harmonic functions) in R n ∖ Λ 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 R + 2 to p-harmonic functions in R n ∖ Λ k when n − k < p < ∞ . Finally, we discuss generalizations of our work to solutions of p -Laplace type PDE (called A -harmonic functions).