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John Lewis: Failure of Fatou type theorems for Solutions to PDE of $p$-Laplace Type in Domains with Flat Boundaries $ \subset \rn{n} $ and in the unit disk of $ \rn{2} $

Time: Thu 2022-12-01 14.00 - 15.00

Location: Institut Mittag-Leffler, Seminar Hall Kuskvillan and Zoom

Video link: Meeting ID: 921 756 1880

Participating: John Lewis - University of Kentucky


Let R^n denote Euclidean n space and 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 discuss the Martin boundary problem for solutions to the p-Laplace equation (called p-harmonic functions) in Rn \ Λk and in R^n_+ relative to {0}. We then indicate how the results from this discussion can be used to extend the work of Tom 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 < ∞ and to the unit disk in R^2+. Finally, time permitting we outline further generalizations of Tom’s work to solutions of p-Laplace type PDE (called A-harmonic functions).