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

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:269

Article

Bjorn, Anders¹  Bjorn, Jana¹  Lehrback, Juha² 

[1] Linkoping Univ, Dept Math, SE-58183 Linkoping, Sweden

[2] Univ Jyvaskyla, Dept Math & Stat, POB 35 MaD, FI-40014 Jyvaskyla, Finland

关键词: Capacitary potential;    Doubling measure;    Metric space;    p-harmonic Green function;    Poincar? inequality;    Singular function;   

DOI  :  10.1016/j.jde.2020.04.044

来源: Elsevier

【 摘 要 】

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 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? inequality, or under similar local assumptions.

【 授权许可】

Free