The Lax-Oleinik semigroup on graphs

Renato Iturriaga; Héctor Sánchez Morgado; Renato Iturriaga; Héctor Sánchez Morgado

doi:10.3934/nhm.2017026

Networks and Heterogeneous Media

2017, Volume 12, Issue 4: 643-662. doi: 10.3934/nhm.2017026

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The Lax-Oleinik semigroup on graphs

Renato Iturriaga ^{1
,},
Héctor Sánchez Morgado ^{2
,
,}

1.
CIMAT, A.P. 402 C.P. 3600, Guanajuato. Gto, México
2.
Instituto de Matemáticas, Universidad Nacional Autónoma de México, Cd. de México C. P. 04510, México

Received: 01 June 2017 Revised: 01 August 2017
Primary: 35R02, 35F21; Secondary: 35Q93

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.
- Lax-Oleinik semigroup,
- weak KAM solution,
- viscosity solution
Citation: Renato Iturriaga, Héctor Sánchez Morgado. The Lax-Oleinik semigroup on graphs[J]. Networks and Heterogeneous Media, 2017, 12(4): 643-662. doi: 10.3934/nhm.2017026

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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.

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