A1 Journal article (refereed)
Existence and almost uniqueness for p-harmonic Green functions on bounded domains in metric spaces (2020)

Björn, A., Björn, J., & Lehrbäck, J. (2020). Existence and almost uniqueness for p-harmonic Green functions on bounded domains in metric spaces. Journal of Differential Equations, 269(9), 6602-6640. https://doi.org/10.1016/j.jde.2020.04.044

JYU authors or editors

Publication details

All authors or editors: Björn, Anders; Björn, Jana; Lehrbäck, Juha

Journal or series: Journal of Differential Equations

ISSN: 0022-0396

eISSN: 1090-2732

Publication year: 2020

Volume: 269

Issue number: 9

Pages range: 6602-6640

Publisher: Elsevier

Publication country: Netherlands

Publication language: English

DOI: https://doi.org/10.1016/j.jde.2020.04.044

Publication open access: Openly available

Publication channel open access: Partially open access channel

Publication is parallel published (JYX): https://jyx.jyu.fi/handle/123456789/69879

Web address of parallel published publication (pre-print): https://arxiv.org/abs/1906.09863

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 Rn, 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.

Keywords: potential theory; metric spaces

Free keywords: capacitary potential; doubling measure; metric space; p-harmonic; green function; Poincaré inequality; singular function

Contributing organizations

Ministry reporting: Yes

Reporting Year: 2020

JUFO rating: 2