On C^0-variational solutions for Hamilton-Jacobi equations

TL;DR

This paper refines the concept of C^0-variational solutions for Hamilton-Jacobi equations, investigates their Markovian properties, and compares variational and viscosity solutions in convex and non-convex cases.

Contribution

It introduces a new definition of C^0-variational solutions, analyzes their Markovian behavior, and provides explicit examples and estimates for different types of Hamiltonians.

Findings

In convex cases, variational solutions are Markovian and coincide with viscosity solutions.

In non-convex cases, minmax and viscous solutions differ, showing limitations of the variational approach.

Upper and lower Hopf-type estimates are established for variational solutions with general initial data.

Abstract

For evolutive Hamilton-Jacobi equations, we propose a refined definition of C^0-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.