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Mathematische Annalen

, Volume 374, Issue 1–2, pp 881–906 | Cite as

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

  • Estibalitz Durand-Cartagena
  • Jesús A. Jaramillo
  • Nageswari ShanmugalingamEmail author
Article
  • 90 Downloads

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 \( p \ge 1\).

Mathematics Subject Classification

Primary 31E05 Secondary 31C45 31C05 54C20 

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Departamento de Matemática Aplicada, ETSI IndustrialesUniversidad Nacional de Educación a Distancia (UNED)MadridSpain
  2. 2.Instituto de Matemática Interdisciplinar and Departamento de Análisis Matemático y Matemática Aplicada, Facultad de CC. MatemáticasUniversidad Complutense de MadridMadridSpain
  3. 3.Department of Mathematical SciencesUniversity of CincinnatiCincinnatiUSA

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