Boundary regularity for $p$-harmonic functions and solutions of obstacle problems on unbounded sets in metric spaces

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, $1<p<\infty$. 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.

arXiv comment21 pages

References in corpus (9)

• Obstacle and Dirichlet problems on arbitrary nonopen sets, and fine topology
• The Dirichlet problem for p-harmonic functions with respect to the Mazurkiewicz boundary
• The Cartan, Choquet and Kellogg properties for the fine topology on metric spaces
• The obstacle and Dirichlet problems associated with p-harmonic functions in unbounded sets in Rn and metric spaces
• The Dirichlet problem for p-harmonic functions on the topologist's comb
• Sphericalization and p-harmonic functions on unbounded domains in Ahlfors regular metric spaces
• The Perron method for $p$-harmonic functions in unbounded sets in ${\mathbf{R}}^n$ and metric spaces
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• Regularity of $p(\cdot )$-superharmonic functions, the Kellogg property and semiregular boundary points

Cited by in corpus (2)

• Condenser capacities and capacitary potentials for unbounded sets, and global $p$-harmonic Green functions on metric spaces
• Semiregular and strongly irregular boundary points for $p$-harmonic functions on unbounded sets in metric spaces