Existence and almost uniqueness for $p$-harmonic Green functions on bounded domains in metric spaces

Abstract

We study ($p$-harmonic) singular functions, defined by means of upper gradients, in bounded domains in metric measure spaces. It is shown that singular functions exist if and only if the complement of the domain has positive capacity, and that they satisfy very precise capacitary identities for superlevel sets. Suitably normalized singular functions are called Green functions. Uniqueness of Green functions is largely an open problem beyond unweighted ${\mathbf{R}}^n$, but we show that all Green functions (in a given domain and with the same singularity) are comparable. As a consequence, for $p$-harmonic functions with a given pole we obtain a similar comparison result near the pole. Various characterizations of singular functions are also given. Our results hold in complete metric spaces with a doubling measure supporting a $p$-Poincaré inequality, or under similar local assumptions.

arXiv comment34 pages

References in corpus (7)

• Obstacle and Dirichlet problems on arbitrary nonopen sets, and fine topology
• Local and semilocal Poincaré inequalities on metric spaces
• The Cartan, Choquet and Kellogg properties for the fine topology on metric spaces
• Sharp capacity estimates for annuli in weighted R^n and in metric spaces
• The variational capacity with respect to nonopen sets in metric spaces
• Local behavior of p-harmonic Green's functions in metric spaces
• Dichotomy of global capacity density in metric measure spaces

Cited by in corpus (5)

• Volume growth, capacity estimates, $p$-parabolicity and sharp integrability properties of $p$-harmonic Green functions
• The convergence rate of $p$-harmonic to infinity-harmonic functions
• Uniqueness and nonuniqueness of $p$-harmonic Green functions on weighted ${\mathbf{R}}^n$ and metric spaces
• Condenser capacities and capacitary potentials for unbounded sets, and global $p$-harmonic Green functions on metric spaces
• Metric Sobolev spaces I: equivalence of definitions