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

Abstract

We construct an explicit representation of viscosity solutions of the Cauchy problem for the Hamilton-Jacobi equation $(H,\sigma )$ on a given domain $\Omega =(0,T)\times {\mathbb{R}}^n.$ It is known that, if the Hamiltonian $H=H(t,p)$ is not a convex (or concave) function in $p$, or $H(\cdot ,p)$ may change its sign on $(0,T)$, then the Hopf-type formula does not define a viscosity solution on $\Omega .$ Under some assumptions for $H(t,p)$ on the subdomains $(t_i,t_{i+1})\times {\mathbb{R}}^n\subset \Omega$, we are able to arrange "partial solutions" given by the Hopf-type formula to get a viscosity solution on $\Omega .$ Then we study the semiconvexity of the solution as well as its relations to characteristics.

arXiv comment15 pages

References in corpus (1)

• Regularity of viscosity solutions defined by Hopf-type formula for Hamilton-Jacobi equations