Sphericalization and p-harmonic functions on unbounded domains in Ahlfors regular metric spaces

Abstract

We use sphericalization to study the Dirichlet problem, Perron solutions and boundary regularity for p-harmonic functions on unbounded sets in Ahlfors regular metric spaces. Boundary regularity for the point at infinity is given special attention. In particular, we allow for several "approach directions" towards infinity and take into account the massiveness of their complements. In 2005, Llorente-Manfredi-Wu showed that the p-harmonic measure on the upper half space $R_{+}^n,n\ge 2$, is not subadditive on null sets when $p\ne 2$. Using their result and spherical inversion, we create similar bounded examples in the unit ball $B\subset R^n$ showing that the n-harmonic measure is not subadditive on null sets when $n\ge 3$, and neither are the p-harmonic measures in $B$ generated by certain weights depending on $p\ne 2$ and $n\ge 2$.

References in corpus (6)

• Prime ends for domains in metric spaces
• The Dirichlet problem for p-harmonic functions with respect to the Mazurkiewicz boundary
• Sharp capacity estimates for annuli in weighted R^n and in metric spaces
• The Mazurkiewicz distance and sets that are finitely connected at the boundary
• The Perron method for $p$-harmonic functions in unbounded sets in ${\mathbf{R}}^n$ and metric spaces
• The Dirichlet problem for p-harmonic functions with respect to arbitrary compactifications

Cited by in corpus (5)

• Classification of metric measure spaces and their ends using $p$-harmonic functions
• Boundary regularity for $p$-harmonic functions and solutions of obstacle problems on unbounded sets in metric spaces
• Condenser capacities and capacitary potentials for unbounded sets, and global $p$-harmonic Green functions on metric spaces
• Preserving Besov (fractional Sobolev) energies under sphericalization and flattening
• Semiregular and strongly irregular boundary points for $p$-harmonic functions on unbounded sets in metric spaces