Article ID Journal Published Year Pages File Type
4616270 Journal of Mathematical Analysis and Applications 2014 12 Pages PDF
Abstract

We construct an explicit representation of viscosity solutions of the Cauchy problem for the Hamilton–Jacobi equation (H,σ)(H,σ) on a given domain Ω=(0,T)×RnΩ=(0,T)×Rn. It is known that, if the Hamiltonian H=H(t,p)H=H(t,p) is not a convex (or concave) function in p  , or H(⋅,p)H(⋅,p) may change its sign on (0,T)(0,T), then the Hopf-type formula does not define a viscosity solution on Ω  . Under some assumptions for H(t,p)H(t,p) on the subdomains (ti,ti+1)×Rn⊂Ω(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.

Related Topics
Physical Sciences and Engineering Mathematics Analysis
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