### Abstract

We establish general representation formulas for solutions of Hamilton-Jacobi equations widi convex Hamiltonians. In order to treat representation formulas on general domains, we introduce a notion of ideal boundary similar to the Martin boundary [21] in potential theory. We apply such representation formulas to investigate maximal solutions, in certain classes of functions, of Hamilton-Jacobi equations. Part of the results in this paper has been announced in [22]. Indiana University Mathematics Journal

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

Pages (from-to) | 2159-2183 |

Number of pages | 25 |

Journal | Indiana University Mathematics Journal |

Volume | 56 |

Issue number | 5 |

DOIs | |

Publication status | Published - 2007 |

### Fingerprint

### Keywords

- Aubry sets
- Hamilton-Jacobi equations
- Representation fotmula
- State constraint problem
- Weak KAM theory

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Indiana University Mathematics Journal*,

*56*(5), 2159-2183. https://doi.org/10.1512/iumj.2007.56.3048

**Representation formulas for solutions of Hamilton-Jacobi equations with convex hamiltonians.** / Ishii, Hitoshi; Mitake, Hiroyoshi.

Research output: Contribution to journal › Article

*Indiana University Mathematics Journal*, vol. 56, no. 5, pp. 2159-2183. https://doi.org/10.1512/iumj.2007.56.3048

}

TY - JOUR

T1 - Representation formulas for solutions of Hamilton-Jacobi equations with convex hamiltonians

AU - Ishii, Hitoshi

AU - Mitake, Hiroyoshi

PY - 2007

Y1 - 2007

N2 - We establish general representation formulas for solutions of Hamilton-Jacobi equations widi convex Hamiltonians. In order to treat representation formulas on general domains, we introduce a notion of ideal boundary similar to the Martin boundary [21] in potential theory. We apply such representation formulas to investigate maximal solutions, in certain classes of functions, of Hamilton-Jacobi equations. Part of the results in this paper has been announced in [22]. Indiana University Mathematics Journal

AB - We establish general representation formulas for solutions of Hamilton-Jacobi equations widi convex Hamiltonians. In order to treat representation formulas on general domains, we introduce a notion of ideal boundary similar to the Martin boundary [21] in potential theory. We apply such representation formulas to investigate maximal solutions, in certain classes of functions, of Hamilton-Jacobi equations. Part of the results in this paper has been announced in [22]. Indiana University Mathematics Journal

KW - Aubry sets

KW - Hamilton-Jacobi equations

KW - Representation fotmula

KW - State constraint problem

KW - Weak KAM theory

UR - http://www.scopus.com/inward/record.url?scp=39049118711&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=39049118711&partnerID=8YFLogxK

U2 - 10.1512/iumj.2007.56.3048

DO - 10.1512/iumj.2007.56.3048

M3 - Article

AN - SCOPUS:39049118711

VL - 56

SP - 2159

EP - 2183

JO - Indiana University Mathematics Journal

JF - Indiana University Mathematics Journal

SN - 0022-2518

IS - 5

ER -