Nonlinear elliptic partial differential equations and \(p\)-harmonic functions on graphs. (English) Zbl 1349.35382

Manfredi, Juan J.; Oberman, Adam M.; Sviridov, Alexander P.

Differ. Integral Equ. 28, No. 1-2, 79-102 (2015).

Summary: 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 partial differential equations. The equations include the graph Laplacian, the \(p\)-Laplacian, the Infinity Laplacian, and the Eikonal operator on the graph.

MSC:

35R02	PDEs on graphs and networks (ramified or polygonal spaces)
65N22	Numerical solution of discretized equations for boundary value problems involving PDEs

Keywords:

partial differential equation on graphs; viscosity solution