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The Lax-Oleinik semigroup on graphs

  • * Corresponding author:Morgado Héctor Sánchez

    * Corresponding author:Morgado Héctor Sánchez
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  • 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.

    Mathematics Subject Classification: Primary: 35R02, 35F21; Secondary: 35Q93.


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