Abstract
Abstract The theory of boundary regularity for p-harmonic functions is extended to unbounded open sets in complete metric spaces with a doubling measure supporting a p-Poincaré inequality, 1 < p < ∞. The barrier classification of regular boundary points is established, and it is shown that regularity is a local property of the boundary. We also obtain boundary regularity results for solutions of the obstacle problem on open sets, and characterize regularity further in several other ways.
Highlights
Let Ω ⊂ Rn be a nonempty bounded open set and let f ∈ C(∂Ω)
The theory of boundary regularity for p-harmonic functions is extended to unbounded open sets in complete metric spaces with a doubling measure supporting a p-Poincaré inequality, < p < ∞
The barrier classi cation of regular boundary points is established, and it is shown that regularity is a local property of the boundary
Summary
Let Ω ⊂ Rn be a nonempty bounded open set and let f ∈ C(∂Ω). The Perron method (introduced on R in 1923 by Perron [47] and independently by Remak [48]) provides a unique function Pf that is harmonic in Ω and takes the boundary values f in a weak sense, i.e., Pf is a solution of the Dirichlet problem for the Laplace equation. These results are new even on unweighted Rn. Boundary regularity for p-harmonic functions on Rn was rst studied by Maz ya [45] who obtained the su ciency part of the Wiener criterion in 1970. Granlund–Lindqvist–Martio [28] were the rst to de ne boundary regularity using Perron solutions for p-harmonic functions, p ≠ They studied the case p = n in Rn and obtained the barrier characterization in this case for bounded open sets. Björn–Björn–Li [14] studied Perron solutions and boundary regular for p-harmonic functions on unbounded open sets in Ahlfors regular metric spaces. There is some overlap with the results in this paper, but it is not substantial and here we consider more general metric spaces than in [14]
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