Existence and almost uniqueness for p-harmonic Green functions on bounded domains in metric spaces
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
Published inJournal of Differential Equations
© 2020 The Authors. Published by Elsevier Inc
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.
Publication in research information system
MetadataShow full item record
Additional information about fundingA.B. and J.B. were supported by the Swedish Research Council, grants 2016-03424 and 621-2014-3974, respectively. J.L. was supported by the Academy of Finland, grant 252108.
Showing items with similar title or keywords.
Arroyo, Ángel; Llorente, José G. (American Mathematical Society, 2019)A metric measure space (X, d, t) is said to satisfy the strong annular decay condition if there is a constant C > 0 such that for each x E X and all 0 < r < R. If do., is the distance induced by the co -norm in RN, we ...
Korte, Riikka; Lahti, Panu; Li, Xining; Shanmugalingam, Nageswari (European Mathematical Society Publishing House, 2019)We study two notions of Dirichlet problem associated with BV energy minimizers (also called functions of least gradient) in bounded domains in metric measure spaces whose measure is doubling and supports a (1, 1)-Poincaré ...
Eriksson-Bique, Sylvester; Vähäkangas, Antti V. (American Mathematical Society, 2019)We prove the self-improvement of a pointwise p-Hardy inequality. The proof relies on maximal function techniques and a characterization of the inequality by curves.
Romney, Matthew (Academic Press, 2019)For all n ≥2, we construct a metric space (X, d)and a quasisymmetric mapping f:[0, 1]n→X with the property that f−1 is not absolutely continuous with respect to the Hausdorff n-measure on X. That is, there exists a Borel ...
Ikonen, Toni (Walter de Gruyter GmbH, 2021)We extend the classical Carathéodory extension theorem to quasiconformal Jordan domains (Y,dY). We say that a metric space (Y,dY) is a quasiconformal Jordan domain if the completion Y of (Y,dY) has finite Hausdor 2-measure, ...