## Abstract

The review article of Crandall, Ishii, and Lions [Bull. AMS, 27, No. 1, 1-67 (1992)] devoted to viscosity solutions of first- and second-order partial differential equations contains the exact Lax formula u(x,t) = inf _{y∈Rn}{v(y) + 1/2t∥x-y∥^{2}} (1) for a solution to the Hamilton-Jacobi nonlinear partial differential equation ∂u/∂t + 1/2∥∇u∥^{2} = 0. u|_{t=0} = v, (2) where the Cauchy data v: R^{n} → R are chosen as a function properly convex and semicontinuous from below, ∥·∥ = 〈·,·) is the usual norm in R^{n}, n ∈ Z_{+}, and t ∈ R _{+} is a positive evolution parameter. The article also states that there is no exact proof of the Lax formula (1) based on general properties of the Hamiltonian-Jacobi equation (2). This work presents precisely such an exact proof of the Lax formula (1).

Original language | English (US) |
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Pages (from-to) | 1541-1547 |

Number of pages | 7 |

Journal | Journal of Mathematical Sciences |

Volume | 99 |

Issue number | 5 |

DOIs | |

State | Published - Jan 1 2000 |

## All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Mathematics(all)
- Applied Mathematics