6533b860fe1ef96bd12c3b12
RESEARCH PRODUCT
Existence and almost uniqueness for p -harmonic Green functions on bounded domains in metric spaces
Anders BjörnJuha LehrbäckJana Björnsubject
Pure mathematicsCapacitary potential; Doubling measure; Metric space; p-harmonic Green function; Poincar? inequality; Singular function31C45 (Primary) 30L99 31C15 31E05 35J92 49Q20 (Secondary)Harmonic (mathematics)Mathematical Analysis01 natural sciencesMeasure (mathematics)Domain (mathematical analysis)Mathematics - Analysis of PDEscapacitary potentialMatematisk analysFOS: MathematicsUniqueness0101 mathematicsMathematicsComplement (set theory)p-harmonicApplied Mathematics010102 general mathematicsmetric spacemetriset avaruudet010101 applied mathematicsMetric spacePoincaré inequalityBounded functionMetric (mathematics)doubling measurepotentiaaliteoriasingular functiongreen functionAnalysisAnalysis of PDEs (math.AP)description
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 $p$-Poincar\'e inequality, or under similar local assumptions.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2020-01-01 |