## Abstract

This article is a continuation of [J. Math. Sci., 99, No. 5, 1541-1547 (2000)] devoted to the validity of the Lax formula (cited in the article of Crandall, Ishii, and Lions [Bull. AMS, 27, No. 1, 1-67 (2000)]) 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∥∇∥^{2} = 0, u|_{t=0+} = v, (2) where the Cauchy data v. ℝ^{n} → ℝ are now a function semicontinuous from below, ∥·∥ = 〈·,·〉 is the usual norm in ℝ^{n}, n ∈ ℤ_{+}, and t ∈ ℝ_{+} is a positive evolution parameter. We proved that the Lax formula solves the Cauchy problem (2) at all points x ∈ ℝ^{n}, t ∈ ℝ_{+} fixed save for an exceptional set of points R of the F_{σ} type, having zero Lebesgue measure. In addition, we formulate a similar Lax-type formula without proof for a solution to a new nonlinear equation of the Hamilton-Jacobi-type: ∂u/∂t + 1/2∥∇u∥^{2} - βu/ 2∥x∥^{2} + 1/2〈Jx, x〉 = 0, where J: ℝ^{n} → ℝ^{n} is a diagonal positive-definite matrix, mentioned in Part I and having interesting applications in modern mathematical physics.

Original language | English (US) |
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Pages (from-to) | 1411-1421 |

Number of pages | 11 |

Journal | Journal of Mathematical Sciences |

Volume | 104 |

Issue number | 5 |

DOIs | |

State | Published - 2001 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

- Statistics and Probability
- General Mathematics
- Applied Mathematics