Nonlinear elliptic Partial Differential Equations and p-harmonic   functions on graphs
  In this article we study the well-posedness (uniqueness and existence of solutions) of nonlinear elliptic Partial Differential Equations (PDEs) on a finite graph. These results are obtained using the discrete comparison principle and connectivity properties of the graph. This work is in the spirit of the theory of viscosity solutions for PDEs.   The equations include the graph Laplacian, the $p$-Laplacian, the Infinity Laplacian, the Mean Curvature equation, and the Eikonal operator on the graph. 
Juan J. Manfredi, Adam M. Oberman, Alex P. Svirodov
math.AP