Abstract
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.
Highlights
In the first part of this article we study the Lax-Oleinik semi-group Lt defined by a Tonelli Lagrangian on a graph and prove that for any continuous function u, Ltu + ct converges as t → ∞ where c is the critical value of the Lagrangian
In the second part of this article we prove that, under the assumption that the Lagrangian is symmetric at the vertices, the sets of weak KAM and viscosity solutions of the Hamilton-Jacobi equation defined coincide
Items (1) and (3) of Corollary 3 imply that v is the maximal dominated function that coincides with u on A and u ≤ v on G
Summary
In the first part of this article we study the Lax-Oleinik semi-group Lt defined by a Tonelli Lagrangian on a graph and prove that for any continuous function u, Ltu + ct converges as t → ∞ where c is the critical value of the Lagrangian. In the second part of this article we prove that, under the assumption that the Lagrangian is symmetric at the vertices, the sets of weak KAM and viscosity solutions of the Hamilton-Jacobi equation defined coincide. We consider a graph G without boundary consisting of finite sets of unoriented edges I = {Ij} and vertices V = {el}. We let Te+l Ij (Te−l Ij) to be the set of Ij- outgoing (incoming or zero) vectors in TelIj
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