On the Lax solution to the Hamilton-Jacobi equation and its generalizations. Part II

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. ℝⁿ → ℝ are now a function semicontinuous from below, ∥·∥ = 〈·,·〉 is the usual norm in ℝⁿ, 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 ∈ ℝⁿ, 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: ℝⁿ → ℝⁿ is a diagonal positive-definite matrix, mentioned in Part I and having interesting applications in modern mathematical physics.