Failure of Fatou type theorems for solutions to PDE of p-Laplace type in domains with flat boundaries. Issue 7 (3rd July 2022)

• Record Type: Journal Article
• Authors: Akman, Murat; Lewis, John; Vogel, Andrew
• Is Part Of: Communications in partial differential equations. Volume 47:Issue 7(2022)
• Page Start: 1457
• Page End: 1503
• Publication Date: 2022-07-03
• Subjects: Gap series -- p-harmonic measure -- p-harmonic function -- radial limits -- Fatou theorem; 35J92 -- 35J60 -- 31B15 -- 39B62 -- 35J20; Differential equations, Partial -- Periodicals; 515.353
• DOI: 10.1080/03605302.2022.2056704
• Languages: English
• ISSNs: 0360-5302

Abstract

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).