Durand Cartagena, EstibalitzJaramillo, Jesús A.2024-12-032024-12-032018-08-22Durand-Cartagena, E., Jaramillo, J.A. & Shanmugalingam, N. Existence and uniqueness of ∞-harmonic functions under assumption of ∞-Poincaré inequality. Math. Ann. 374, 881–906 (2019). https://doi.org/10.1007/s00208-018-1747-z1432-1807https://doi.org/10.1007/s00208-018-1747-zhttps://hdl.handle.net/20.500.14468/24672The registered version of this article, first published in Mathematische Annalen, is available online at the publisher's website: Springer Nature, https://doi.org/10.1007/s00208-018-1747-zLa versión registrada de este artículo, publicado por primera vez en Mathematische Annalen, está disponible en línea en el sitio web del editor: Springer Nature, https://doi.org/10.1007/s00208-018-1747-zGiven a complete metric measure space whose measure is doubling and supports an ∞- Poincar´e inequality, and a bounded domain Ω in such a space together with a Lipschitz function f : ∂Ω → R, we show the existence and uniqueness of an ∞-harmonic extension of f to Ω. To do so, we show that there is a metric that is bi-Lipschitz equivalent to the original metric, such that with respect to this new metric the metric space satisfies an ∞- weak Fubini property and that a function which is ∞-harmonic in the original metric must also be ∞-harmonic with respect to the new metric. We also show that if the metric on the metric space satisfies an ∞-weak Fubini property, then the notion of ∞-harmonic functions coincide with the notion of AMLEs proposed by Aronsson. The notion of ∞-harmonicity is in general distinct from the notion of strongly absolutely minimizing Lipschitz extensions found in [13, 25, 26], but coincides when the metric space supports a p-Poincar´e inequality for some finite p ≥ 1.eninfo:eu-repo/semantics/openAccess12 Matemáticasartículo∞-Poincar´e inequality∞-harmonicAMLEmetric measure spaces