Abstract
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.
Highlights
We study some properties of a measure implying its continuity with respect to the given metric and notice that this condition gives us wide class of metric measure spaces
Our studies involve various other types of measures, e.g. uniform measures and measures satisfying δ-annular decay condition for some δ ∈
For strongly harmonic functions we prove this result for geodesic doubling measure spaces with the Hölder exponent depending on the doubling constant only, whereas for weakly harmonic functions we, require that a compact set K remains enough away from the boundary of the domain and the admissible radii for points in K are uniformly separated from zero and uniformly bounded from the above
Summary
We define continuity of a measure with respect to a metric, see Definition 2.1 Such a property has been important in the previous studies of harmonic functions, see [13] ( [15]). 3 we bring on stage main characters of the paper, i.e. strongly and weakly harmonic functions, motivate their definitions and introduce some of their basic properties and natural relatives such as sub- and superharmonic functions The latter two notions will play a vital role in the studies of the Dirichlet problem in Sect. In Theorem 7.1 we provide a fairly general condition for a measure which implies that a strongly(weakly) bounded harmonic function defined in the whole space must be constant. For instance, show that even in a simple case of R there exist non-Lebesgue measures for which bounded entire harmonic functions need not be constant
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