Abstract
AbstractIn this paper, we study the relationship between p ‐harmonic functions and absolutely minimizing Lipschitz extensions in the setting of a metric measure space (X, d, μ ). In particular, we show that limits of p ‐harmonic functions (as p → ∞) are necessarily the∞ ‐energy minimizers among the class of all Lipschitz functions with the same boundary data. Our research is motivated by the observation that while the p ‐harmonic functions in general depend on the underlying measure μ , in many cases their asymptotic limit as p → ∞ turns out have a characterization that is independent of the measure. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
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