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

TL;DR

This paper develops a method to construct viscosity solutions for nonautonomous Hamilton-Jacobi equations with non-convex Hamiltonians, analyzing their semiconvexity and characteristic relations on complex domains.

Contribution

It introduces a layered approach to build viscosity solutions from partial Hopf-type solutions for non-convex Hamiltonians on subdivided domains.

Findings

Explicit layered representation of viscosity solutions.

Conditions under which solutions maintain semiconvexity.

Relations between solutions and characteristics are established.

Abstract

We construct an explicit representation of viscosity solutions of the Cauchy problem for the Hamilton-Jacobi equation $(H,\sigma)$ on a given domain $\Omega= (0,T)\times \R^n.$ 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 $(t_i, t_{i+1})\times \R^n\subset \Omega$, 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.