Harmonic functions on metric measure spaces

Tomasz Adamowicz , Michał Gaczkowski , Przemysław Górka


We introduce and study strongly and weakly harmonic functions on metric measure spaces defined via the mean value property holding for all and, respectively, for some radii of balls at every point of the underlying domain. Among properties of such functions we investigate various types of Harnack estimates on balls and compact sets, weak and strong maximum principles, comparison principles, the Hölder and the Lipschitz estimates and some differentiability properties. The latter one is based on the notion of a weak upper gradient. The Dirichlet problem for functions satisfying the mean value property is studied via the dynamical programming method related to stochastic games. Finally, we discuss and prove the Liouville type theorems. Our results are obtained for various types of measures: continuous with respect to a metric, doubling, uniform, measures satisfying the annular decay condition. Relations between such measures are presented as well. The presentation is illustrated by examples.
Author Tomasz Adamowicz - [Institute of Mathematics (IM PAN) [Polska Akademia Nauk (PAN)]]
Tomasz Adamowicz,,
- Instytut Matematyczny
, Michał Gaczkowski (FMIS / DFE)
Michał Gaczkowski,,
- Department of Functional Equations
, Przemysław Górka (FMIS / DPDE)
Przemysław Górka,,
- Department of Partial Differential Equations
Journal seriesRevista Matematica Complutense, ISSN 1139-1138, (A 20 pkt)
Issue year2019
Publication size in sheets2.25
Keywords in EnglishDirichlet problem, Doubling measure, Dynamical programming, Harmonic function, Harnack estimate, Hölder continuity, Liouville theorem, Lipschitz continuity, Mean value property, Measure, Metric analysis, Potential theory, Uniform measure, Weak upper gradient
ASJC Classification2600 General Mathematics
URL https://link.springer.com/article/10.1007/s13163-018-0272-7
Languageen angielski
Score (nominal)20
ScoreMinisterial score = 20.0, 28-05-2019, ArticleFromJournal
Publication indicators WoS Citations = 0; Scopus Citations = 1; Scopus SNIP (Source Normalised Impact per Paper): 2016 = 1.43; WoS Impact Factor: 2017 = 1.055 (2) - 2017=0.844 (5)
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* presented citation count is obtained through Internet information analysis and it is close to the number calculated by the Publish or Perish system.