# American Institute of Mathematical Sciences

April  2014, 34(4): 1683-1700. doi: 10.3934/dcds.2014.34.1683

## On landscape functions associated with transport paths

 1 University of California at Davis, Department of Mathematics, One Shields Ave, Davis,CA, 95616, United States

Received  October 2012 Revised  March 2013 Published  October 2013

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.
Citation: Qinglan Xia. On landscape functions associated with transport paths. Discrete & Continuous Dynamical Systems, 2014, 34 (4) : 1683-1700. doi: 10.3934/dcds.2014.34.1683
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