On the lax solution to the Hamilton-Jacobi equation. Part I

A. K. Prykarpatsky, D. L. Blackmore

Research output: Contribution to journalReview articlepeer-review

1 Scopus citations

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: Rn → R are chosen as a function properly convex and semicontinuous from below, ∥·∥ = 〈·,·) is the usual norm in Rn, 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 languageEnglish (US)
Pages (from-to)1541-1547
Number of pages7
JournalJournal of Mathematical Sciences
Volume99
Issue number5
DOIs
StatePublished - Jan 1 2000

All Science Journal Classification (ASJC) codes

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

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