American Institute of Mathematical Sciences

Advanced
October  2007, 17(4): 807-819. doi: 10.3934/dcds.2007.17.807

A note on singular perturbation problems via Aubry-Mather theory

Fabio Camilli 1, and  Annalisa Cesaroni 2,

1. 

Dip. di Matematica Pura e Applicata, Univ. dell’Aquila, loc. Monteluco di Roio, 67040 l’Aquila, Italy
2. 

Dip. di Matematica Pura e Applicata, Univ. di Padova, via Trieste 63, 35131 Padova, Italy

Received  June 2006 Revised  October 2006 Published  January 2007

Exploiting the metric approach to Hamilton-Jacobi equation recently introduced by Fathi and Siconolfi [13], we prove a singular perturbation result for a general class of Hamilton-Jacobi equations. Considered in the framework of small random perturbations of dynamical systems, it extends a result due to Kamin [19] to the case of a dynamical system having several attracting points inside the domain.
Keywords: PDE Aubry-Mather theory., Singular perturbations, viscosity solutions.
Mathematics Subject Classification: Primary: 35B25 ; Secondary: 49L25, 35F30, 37J5.
Citation: Fabio Camilli, Annalisa Cesaroni. A note on singular perturbation problems via Aubry-Mather theory. Discrete & Continuous Dynamical Systems - A, 2007, 17 (4) : 807-819. doi: 10.3934/dcds.2007.17.807
[1]

Ugo Bessi. Viscous Aubry-Mather theory and the Vlasov equation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 379-420. doi: 10.3934/dcds.2014.34.379
[2]

Hans Koch, Rafael De La Llave, Charles Radin. Aubry-Mather theory for functions on lattices. Discrete & Continuous Dynamical Systems - A, 1997, 3 (1) : 135-151. doi: 10.3934/dcds.1997.3.135
[3]

Diogo A. Gomes. Viscosity solution methods and the discrete Aubry-Mather problem. Discrete & Continuous Dynamical Systems - A, 2005, 13 (1) : 103-116. doi: 10.3934/dcds.2005.13.103
[4]

Yasuhiro Fujita, Katsushi Ohmori. Inequalities and the Aubry-Mather theory of Hamilton-Jacobi equations. Communications on Pure & Applied Analysis, 2009, 8 (2) : 683-688. doi: 10.3934/cpaa.2009.8.683
[5]

Artur O. Lopes, Rafael O. Ruggiero. Large deviations and Aubry-Mather measures supported in nonhyperbolic closed geodesics. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 1155-1174. doi: 10.3934/dcds.2011.29.1155
[6]

Bassam Fayad. Discrete and continuous spectra on laminations over Aubry-Mather sets. Discrete & Continuous Dynamical Systems - A, 2008, 21 (3) : 823-834. doi: 10.3934/dcds.2008.21.823
[7]

Siniša Slijepčević. The Aubry-Mather theorem for driven generalized elastic chains. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2983-3011. doi: 10.3934/dcds.2014.34.2983
[8]

Mads R. Bisgaard. Mather theory and symplectic rigidity. Journal of Modern Dynamics, 2019, 15: 165-207. doi: 10.3934/jmd.2019018
[9]

Ogabi Chokri. On the $L^p-$ theory of Anisotropic singular perturbations of elliptic problems. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1157-1178. doi: 10.3934/cpaa.2016.15.1157
[10]

Youri V. Egorov, Evariste Sanchez-Palencia. Remarks on certain singular perturbations with ill-posed limit in shell theory and elasticity. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1293-1305. doi: 10.3934/dcds.2011.31.1293
[11]

Christian Lax, Sebastian Walcher. Singular perturbations and scaling. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 1-29. doi: 10.3934/dcdsb.2019170
[12]

Diego Castellaneta, Alberto Farina, Enrico Valdinoci. A pointwise gradient estimate for solutions of singular and degenerate pde's in possibly unbounded domains with nonnegative mean curvature. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1983-2003. doi: 10.3934/cpaa.2012.11.1983
[13]

Zvi Artstein. Invariance principle in the singular perturbations limit. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3653-3666. doi: 10.3934/dcdsb.2018309
[14]

Kung-Ching Chang, Xuefeng Wang, Xie Wu. On the spectral theory of positive operators and PDE applications. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3171-3200. doi: 10.3934/dcds.2020054
[15]

Michel Chipot, Senoussi Guesmia. On the asymptotic behavior of elliptic, anisotropic singular perturbations problems. Communications on Pure & Applied Analysis, 2009, 8 (1) : 179-193. doi: 10.3934/cpaa.2009.8.179
[16]

Senoussi Guesmia, Abdelmouhcene Sengouga. Some singular perturbations results for semilinear hyperbolic problems. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 567-580. doi: 10.3934/dcdss.2012.5.567
[17]

Claudio Marchi. On the convergence of singular perturbations of Hamilton-Jacobi equations. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1363-1377. doi: 10.3934/cpaa.2010.9.1363
[18]

Chiara Zanini. Singular perturbations of finite dimensional gradient flows. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 657-675. doi: 10.3934/dcds.2007.18.657
[19]

Canela Jordi. Singular perturbations of Blaschke products and connectivity of Fatou components. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3567-3585. doi: 10.3934/dcds.2017153
[20]

Manfred Deistler. Singular arma systems: A structure theory. Numerical Algebra, Control & Optimization, 2019, 9 (3) : 383-391. doi: 10.3934/naco.2019025

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (34)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences