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A multiscale model reduction method for nonlinear monotone elliptic equations in heterogeneous media
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 |
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 [
References:
| [1] |
Y. Achdou, F. Camilli, A. 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. Barles, A. 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. Achdou, F. Camilli, A. 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. Barles, A. 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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