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} $

Geometric Aspects of Nonlinear Partial Differential Equations

01 December 14:00 - 15:00

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).
Panagiota Daskalopoulos
Columbia University
Alessio Figalli
ETH Zürich
Erik Lindgren
Uppsala University
Henrik Shahgholian
KTH Royal Institute of Technology
Susanna Terracini,
University of Turin


Erik Lindgren

Henrik Shahgholian


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