### Abstract

The equations u_{t} + H(Du) = 0 and u_{t} + H(u, Du) = 0, with initial condition u(0, x) = g(x) have an explicit solution when the hamiltonian is convex in the gradient variable (Lax formula) or the initial data is convex, or quasiconvex (Hopf formula). This paper extends these formulas to initial functions g which are only lower semicontinuous (lsc), and possibly infinite. It is proved that the Lax formulas give a lsc viscosity solution, and the Hopf formulas result in the minimal supersolution. A level set approach is used to give the most general results.

Original language | English |
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Pages (from-to) | 993-1035 |

Number of pages | 43 |

Journal | Indiana University Mathematics Journal |

Volume | 48 |

Issue number | 3 |

Publication status | Published - 1999 Sep |

Externally published | Yes |

### Keywords

- Hopf and Lax formulas
- Level sets
- lsc viscosity solutions

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

Alvarez, O., Barron, E. N., & Ishii, H. (1999). Hopf-Lax Formulas for Semicontinuous Data.

*Indiana University Mathematics Journal*,*48*(3), 993-1035.