December 2017, 12(4): 643-662. doi: 10.3934/nhm.2017026

The Lax-Oleinik semigroup on graphs

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

* Corresponding author:Morgado Héctor Sánchez

Received  June 2017 Revised  August 2017 Published  October 2017

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.

Citation: Renato Iturriaga, Héctor Sánchez Morgado. The Lax-Oleinik semigroup on graphs. Networks & Heterogeneous Media, 2017, 12 (4) : 643-662. doi: 10.3934/nhm.2017026
References:
[1]

Y. AchdouF. CamilliA. Cutrí and N. Tchou, Hamilton-Jacobi equations constrained on networks, Nonlinear Differ. Equ. Appl., 20 (2013), 413-445. doi: 10.1007/s00030-012-0158-1.

[2]

G. BarlesA. Briani and E. Chasseigne, A Bellman approach for two-domains optimal control problems in ${\mathbb{R}^n}$, ESAIM: Control, Optimisation and Calculus of Variations, 19 (2013), 710-739. doi: 10.1051/cocv/2012030.

[3]

F. Camilli and D. Schieborn, Viscosity solutions of Eikonal equations on topological networks, Calc. Var. Partial Differential Equatons, 46 (2013), 671-686. doi: 10.1007/s00526-012-0498-z.

[4]

F. Camilli and C. Marchi, A comparison among various notions of viscosity solution for Hamilton-Jacobi equations on networks, J. Math. Anal. Appl., 407 (2013), 112-118. doi: 10.1016/j.jmaa.2013.05.015.

[5]

A. Davini and A. Siconolfi, A generalized dynamical approach to the large time behavior of solutions of Hamilton-Jacobi equations, SIAM J. Math. Anal., 38 (2006), 478-502. doi: 10.1137/050621955.

[6]

A. Fathi, Sur la convergence du semi-groupe de Lax-Oleinik, C. R. Acad. Sci. Paris Sr. I Math., 327 (1998), 267-270. doi: 10.1016/S0764-4442(98)80144-4.

[7]

A. Fathi, Weak KAM Theorem in Lagrangian Dynamics, To appear in Cambridge Studies in Advanced Mathematics.

[8]

A. Fathi and A. Siconolfi, PDE aspects of Aubry-Mather theory for quasi-convex Hamiltonians, Calc. Var. Partial Differential Equations, 22 (2005), 185-228. doi: 10.1007/s00526-004-0271-z.

[9]

C. Imbert and R. Monneau, Flux-limited Solutions for Quasi-Convex Hamilton-Jacobi Equations on Networks, arXiv: 1306.2428

[10]

H. Ishii, Asymptotic solutions for large time of Hamilton-Jacobi equations in Euclidean nspace, Anal. Non Linéaire, 25 (2008), 231-266. doi: 10.1016/j.anihpc.2006.09.002.

[11]

J. M. Roquejoffre, Convergence to steady states or periodic solutions in a class of HamiltonJacobi equations, J. Math. Pures Appl., 80 (2001), 85-104. doi: 10.1016/S0021-7824(00)01183-1.

show all references

References:
[1]

Y. AchdouF. CamilliA. Cutrí and N. Tchou, Hamilton-Jacobi equations constrained on networks, Nonlinear Differ. Equ. Appl., 20 (2013), 413-445. doi: 10.1007/s00030-012-0158-1.

[2]

G. BarlesA. Briani and E. Chasseigne, A Bellman approach for two-domains optimal control problems in ${\mathbb{R}^n}$, ESAIM: Control, Optimisation and Calculus of Variations, 19 (2013), 710-739. doi: 10.1051/cocv/2012030.

[3]

F. Camilli and D. Schieborn, Viscosity solutions of Eikonal equations on topological networks, Calc. Var. Partial Differential Equatons, 46 (2013), 671-686. doi: 10.1007/s00526-012-0498-z.

[4]

F. Camilli and C. Marchi, A comparison among various notions of viscosity solution for Hamilton-Jacobi equations on networks, J. Math. Anal. Appl., 407 (2013), 112-118. doi: 10.1016/j.jmaa.2013.05.015.

[5]

A. Davini and A. Siconolfi, A generalized dynamical approach to the large time behavior of solutions of Hamilton-Jacobi equations, SIAM J. Math. Anal., 38 (2006), 478-502. doi: 10.1137/050621955.

[6]

A. Fathi, Sur la convergence du semi-groupe de Lax-Oleinik, C. R. Acad. Sci. Paris Sr. I Math., 327 (1998), 267-270. doi: 10.1016/S0764-4442(98)80144-4.

[7]

A. Fathi, Weak KAM Theorem in Lagrangian Dynamics, To appear in Cambridge Studies in Advanced Mathematics.

[8]

A. Fathi and A. Siconolfi, PDE aspects of Aubry-Mather theory for quasi-convex Hamiltonians, Calc. Var. Partial Differential Equations, 22 (2005), 185-228. doi: 10.1007/s00526-004-0271-z.

[9]

C. Imbert and R. Monneau, Flux-limited Solutions for Quasi-Convex Hamilton-Jacobi Equations on Networks, arXiv: 1306.2428

[10]

H. Ishii, Asymptotic solutions for large time of Hamilton-Jacobi equations in Euclidean nspace, Anal. Non Linéaire, 25 (2008), 231-266. doi: 10.1016/j.anihpc.2006.09.002.

[11]

J. M. Roquejoffre, Convergence to steady states or periodic solutions in a class of HamiltonJacobi equations, J. Math. Pures Appl., 80 (2001), 85-104. doi: 10.1016/S0021-7824(00)01183-1.

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