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Existence and uniqueness of \(\infty \)-harmonic functions under assumption of \(\infty \)-Poincaré inequality

Abstract

Given a complete metric measure space whose measure is doubling and supports an \(\infty \)-Poincaré inequality, and a bounded domain \(\Omega \) in such a space together with a Lipschitz function \(f:\partial \Omega \rightarrow {\mathbb {R}}\), we show the existence and uniqueness of an \(\infty \)-harmonic extension of f to \(\Omega \). 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 \(\infty \)-weak Fubini property and that a function which is \(\infty \)-harmonic in the original metric must also be \(\infty \)-harmonic with respect to the new metric. We also show that if the metric on the metric space satisfies an \(\infty \)-weak Fubini property, then the notion of \(\infty \)-harmonic functions coincide with the notion of AMLEs proposed by Aronsson. The notion of \(\infty \)-harmonicity is in general distinct from the notion of strongly absolutely minimizing Lipschitz extensions found in Crandall et al. (Calc Var Partial Differ Equ 13: 123–139, 2001), Juutinen (Ann Acad Sci Fenn Math 27(1):57–67, 2002), Juutinen and Shanmugalingam (Math Nachr 279(9–10):1083–1098, 2006), but coincides when the metric space supports a p-Poincaré inequality for some finite \( p \ge 1\).

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Correspondence to Nageswari Shanmugalingam.

Additional information

N.S.’s research was partially supported from Grant # DMS-1500440 of NSF (U.S.A.). The research of J.J. and E.D-C. are partially supported by Grant MTM2015-65825-P (Spain). E.D-C. was also partially supported from 2016-MAT09 and 2018-MAT14 (Apoyo Investigación Matemática Aplicada, ETSI Industriales, UNED). Part of the research was conducted during N.S.’s visit to ICMAT (Madrid, Spain) in Spring 2015 and to the Universidad Complutense de Madrid in Spring 2018, and during the visit of E.D-C. and J.J. to the University of Cincinnati in Spring 2016. The authors thank these institutions for their kind hospitality. The authors also wish to thank Riikka Korte for interesting discussions regarding moduli of curve families, and the anonymous referees for suggestions that helped to improve the exposition of the paper and correct a mathematical error.

Communicated by Y. Giga.

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Durand-Cartagena, E., Jaramillo, J.A. & Shanmugalingam, N. Existence and uniqueness of \(\infty \)-harmonic functions under assumption of \(\infty \)-Poincaré inequality. Math. Ann. 374, 881–906 (2019). https://doi.org/10.1007/s00208-018-1747-z

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Mathematics Subject Classification

  • Primary 31E05
  • Secondary 31C45
  • 31C05
  • 54C20