2009/04/29 by Olga Bernardi, Bernardi, Olga, Franco Cardin +1
Computer Science · Mathematics · #35A30 #35D99 #53D35 #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Optimization and Variational Analysis #Symplectic Geometry (math.SG) #math.AP #math.SG #msc:35A30 #msc:35D99 #msc:53D35
paper · pdf · doi:10.48550/arxiv.0904.4557
30 pages
arxiv created 2009/04/29 · openalex publication_date 2009/04/29 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For evolutive Hamilton-Jacobi equations, we propose a refined definition of C0-variational solution, adapted to Cauchy problems for continuous initial data. In this weaker framework we investigate the Markovian (or semigroup) property for these solutions. In the case of p-convex Hamiltonians, when variational solutions are known to be identical to viscosity solutions, we verify directly the Markovian property by using minmax techniques. In the non-convex case, we construct an explicit evolutive example where minmax and viscous solutions are different. Provided the initial data allow for the separation of variables, we also detect the Markovian property for convex-concave Hamiltonians. In this case, and for general initial data, we finally give upper and lower Hopf-type estimates for the variational solutions.