### Abstract

We show firstly the equivalence between existence of a periodic solution of the Hamilton-Jacobi equation 'formula presented' is a bounded domain of 'formula presented', with the Dirichlet boundary condition 'formula presented' and that of a subsolution of the stationary problem 'formula presented' under the assumptions that the function 'formula presented' is periodic in t and H is coercive. Here 'formula presented' denotes the average of f over the period. This proposition is a variant of a recent result for 'formula presented' due to Bostan-Namah, and we give a different and simpler approach to such an equivalence. Secondly, we establish that any periodic solution u(x, t) of the problem, ut + H(x, Du) = 0 in 'formula presented' and 'formula presented', is constant in t on the Aubry set for H. Here H is assumed to be convex, coercive and strictly convex in a sense.

Original language | English |
---|---|

Title of host publication | Recent Progress on Reaction-Diffusion Systems and Viscosity Solutions |

Publisher | World Scientific Publishing Co. |

Pages | 97-119 |

Number of pages | 23 |

ISBN (Print) | 9789812834744, 9812834737, 9789812834737 |

DOIs | |

Publication status | Published - 2009 Jan 1 |

### Fingerprint

### Keywords

- Aubry sets
- Hamilton-Jacobi equations
- Periodic solutions

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Recent Progress on Reaction-Diffusion Systems and Viscosity Solutions*(pp. 97-119). World Scientific Publishing Co.. https://doi.org/10.1142/9789812834744_0005

**Two remarks on periodic solutions of Hamilton-Jacobi equations.** / Ishii, Hitoshi; Mitake, Hiroyoshi.

Research output: Chapter in Book/Report/Conference proceeding › Chapter

*Recent Progress on Reaction-Diffusion Systems and Viscosity Solutions.*World Scientific Publishing Co., pp. 97-119. https://doi.org/10.1142/9789812834744_0005

}

TY - CHAP

T1 - Two remarks on periodic solutions of Hamilton-Jacobi equations

AU - Ishii, Hitoshi

AU - Mitake, Hiroyoshi

PY - 2009/1/1

Y1 - 2009/1/1

N2 - We show firstly the equivalence between existence of a periodic solution of the Hamilton-Jacobi equation 'formula presented' is a bounded domain of 'formula presented', with the Dirichlet boundary condition 'formula presented' and that of a subsolution of the stationary problem 'formula presented' under the assumptions that the function 'formula presented' is periodic in t and H is coercive. Here 'formula presented' denotes the average of f over the period. This proposition is a variant of a recent result for 'formula presented' due to Bostan-Namah, and we give a different and simpler approach to such an equivalence. Secondly, we establish that any periodic solution u(x, t) of the problem, ut + H(x, Du) = 0 in 'formula presented' and 'formula presented', is constant in t on the Aubry set for H. Here H is assumed to be convex, coercive and strictly convex in a sense.

AB - We show firstly the equivalence between existence of a periodic solution of the Hamilton-Jacobi equation 'formula presented' is a bounded domain of 'formula presented', with the Dirichlet boundary condition 'formula presented' and that of a subsolution of the stationary problem 'formula presented' under the assumptions that the function 'formula presented' is periodic in t and H is coercive. Here 'formula presented' denotes the average of f over the period. This proposition is a variant of a recent result for 'formula presented' due to Bostan-Namah, and we give a different and simpler approach to such an equivalence. Secondly, we establish that any periodic solution u(x, t) of the problem, ut + H(x, Du) = 0 in 'formula presented' and 'formula presented', is constant in t on the Aubry set for H. Here H is assumed to be convex, coercive and strictly convex in a sense.

KW - Aubry sets

KW - Hamilton-Jacobi equations

KW - Periodic solutions

UR - http://www.scopus.com/inward/record.url?scp=84969627637&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84969627637&partnerID=8YFLogxK

U2 - 10.1142/9789812834744_0005

DO - 10.1142/9789812834744_0005

M3 - Chapter

AN - SCOPUS:84969627637

SN - 9789812834744

SN - 9812834737

SN - 9789812834737

SP - 97

EP - 119

BT - Recent Progress on Reaction-Diffusion Systems and Viscosity Solutions

PB - World Scientific Publishing Co.

ER -