### 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 |
---|---|

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 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Indiana University Mathematics Journal*,

*48*(3), 993-1035.

**Hopf-Lax Formulas for Semicontinuous Data.** / Alvarez, O.; Barron, E. N.; Ishii, Hitoshi.

Research output: Contribution to journal › Article

*Indiana University Mathematics Journal*, vol. 48, no. 3, pp. 993-1035.

}

TY - JOUR

T1 - Hopf-Lax Formulas for Semicontinuous Data

AU - Alvarez, O.

AU - Barron, E. N.

AU - Ishii, Hitoshi

PY - 1999/9

Y1 - 1999/9

N2 - The equations ut + H(Du) = 0 and ut + 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.

AB - The equations ut + H(Du) = 0 and ut + 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.

KW - Hopf and Lax formulas

KW - Level sets

KW - lsc viscosity solutions

UR - http://www.scopus.com/inward/record.url?scp=0001262246&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0001262246&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0001262246

VL - 48

SP - 993

EP - 1035

JO - Indiana University Mathematics Journal

JF - Indiana University Mathematics Journal

SN - 0022-2518

IS - 3

ER -