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On landscape functions associated with transport paths

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  • In this paper, we introduce a multiple-sources version of the landscape function which was originally introduced by Santambrogio in [10]. More precisely, we study landscape functions associated with a transport path between two atomic measures of equal mass. We also study p-harmonic functions on a directed graph for nonpositive $p$. We show an equivalence relation between landscape functions associated with an $\alpha $-transport path and $ p$-harmonic functions on the underlying graph of the transport path for $ p=\alpha /(\alpha -1)$, which is the conjugate of $\alpha $. Furthermore, we prove the Lipschitz continuity of a landscape function associated with an optimal transport path on each of its connected components.
    Mathematics Subject Classification: Primary: 90B06, 49Q20.


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