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 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 - 2000 |
All Science Journal Classification (ASJC) codes
- Statistics and Probability
- General Mathematics
- Applied Mathematics