The Lax-Oleinik semigroup on graphs
Renato Iturriaga Héctor Sánchez Morgado
Networks & Heterogeneous Media 2017, 12(4): 643-662 doi: 10.3934/nhm.2017026

We consider Tonelli Lagrangians on a graph, define weak KAM solutions, which happen to be the fixed points of the Lax-Oleinik semi-group, and identify their uniqueness set as the Aubry set, giving a representation formula. Our main result is the long time convergence of the Lax Oleinik semi-group. It follows that weak KAM solutions are viscosity solutions of the Hamilton-Jacobi equation [3, 4], and in the case of Hamiltonians called of eikonal type in [3], we prove that the converse holds.

keywords: Lax-Oleinik semigroup weak KAM solution viscosity solution

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