Abstract
We construct an explicit representation of viscosity solutions of the Cauchy problem for the Hamilton–Jacobi equation (H,σ) on a given domain Ω=(0,T)×Rn. It is known that, if the Hamiltonian H=H(t,p) is not a convex (or concave) function in p, or H(⋅,p) may change its sign on (0,T), then the Hopf-type formula does not define a viscosity solution on Ω. Under some assumptions for H(t,p) on the subdomains (ti,ti+1)×Rn⊂Ω, we are able to arrange “partial solutions” given by the Hopf-type formula to get a viscosity solution on Ω. Then we study the semiconvexity of the solution as well as its relations to characteristics.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
