Layered Viscosity Solutions of Nonautonomous Hamilton-Jacobi Equations:\n Semiconvexity and Relations to Characteristics

2012/04/25 by Nguyen Hoang, Hoang, Nguyen, Nguyen Mau Nam +2
Mathematics · Medicine · #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Mathematical Biology Tumor Growth #Mathematical and Theoretical Epidemiology and Ecology Models #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1204.5628

openalex publication_date 2012/04/25 · openalex created_date 2022/10/10 · openalex updated_date 2026/07/28

Abstract

We construct an explicit representation of viscosity solutions of the Cauchy\nproblem for the Hamilton-Jacobi equation (H,\σ) on a given domain\n\Ω= (0,T)\× Rⁿ. It is known that, if the Hamiltonian H = H(t,p)\nis not a convex (or concave) function in p, or H(\⋅, p) may change its\nsign on (0,T), then the Hopf-type formula does not define a viscosity\nsolution on \Ω. Under some assumptions for H(t,p) on the subdomains\n(t_i, t_i+1)\× Rⁿ\⊂ \Ω, we are able to arrange "partial\nsolutions" given by the Hopf-type formula to get a viscosity solution on\n\Ω. Then we study the semiconvexity of the solution as well as its\nrelations to characteristics.\n

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