Layered viscosity solutions of nonautonomous Hamilton-Jacobi equations: Semiconvexity and relations to characteristics

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:410

Article

Nguyen Hoang¹  Nam, Nguyen Mau² 

[1] Hue Univ, Coll Educ, Dept Math, Hue, Vietnam

[2] Portland State Univ, Fariborz Maseeh Dept Math & Stat, Portland, OR 97207 USA

关键词: Hamilton-Jacobi equation;    Hopf-type formula;    Layered viscosity solution;    Semiconvexity;   

DOI  :  10.1016/j.jmaa.2013.08.047

来源: Elsevier

【 摘 要 】

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) x R-n. 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 Omega. Under some assumptions for H(t, p) on the subdomains (t(i), t(i+1)) x R-n subset of 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. (C) 2013 Elsevier Inc. All rights reserved.

【 授权许可】

Free