Existence and almost uniqueness for pharmonic Green functions on bounded domains in metric spaces
Björn, A., Björn, J., & Lehrbäck, J. (2020). Existence and almost uniqueness for pharmonic Green functions on bounded domains in metric spaces. Journal of Differential Equations, 269(9), 66026640. https://doi.org/10.1016/j.jde.2020.04.044
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Journal of Differential EquationsDate
2020Copyright
© 2020 The Authors. Published by Elsevier Inc
We study (pharmonic) 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 pharmonic 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 pPoincaré inequality, or under similar local assumptions.
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A.B. and J.B. were supported by the Swedish Research Council, grants 201603424 and 62120143974, respectively. J.L. was supported by the Academy of Finland, grant 252108.License
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