On \(C^0\)-variational solutions for Hamilton-Jacobi equations. (English) Zbl 1222.70018
Summary: For evolutive Hamilton-Jacobi equations, we propose a refined definition of \(C^0\)-variational solution, adapted to Cauchy problems for continuous initial data. This weaker framework enables us to investigate the 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 semigroup property by using minmax techniques. In the non-convex case, we construct a first explicit evolutive example where minmax and viscosity solutions are different. Provided the initial data allow for the separation of variables, we also detect the semigroup property for convex-concave Hamiltonians. In this case, and for general initial data, we finally give new upper and lower Hopf-type estimates for the variational solutions.
MSC:
70H20 | Hamilton-Jacobi equations in mechanics |
35D30 | Weak solutions to PDEs |
37J05 | Relations of dynamical systems with symplectic geometry and topology (MSC2010) |