Minimax and viscosity solutions of Hamilton-Jacobi equations in the convex case

Abstract

The aim of this paper is twofold. We construct an extension to a non-integrable case of Hopf's formula, often used to produce viscosity solutions of Hamilton-Jacobi equations for p-convex integrable Hamiltonians. Furthermore, for a general class of p-convex Hamiltonians, we present a proof of the equivalence of the minimax solution with the viscosity solution.

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

“…For the following result we refer to [33], [8] and [7]. As a consequence of the previous Proposition 3.2, Theorem 2.3 guarantees that the Lagrangian submanifold L admits essentially (that is, up to the three operations described above) an unique GFQI…”

“…The reader is referred to [20], [5], [3] for general review on the theory. Even thought variational and viscosity solutions have the same analytic properties, it is not known in the general non-convex case whether they coincide or not, although they do for p-convex Hamiltonians ( [16], [7]). In Section 4.1 we construct an evolutive example showing explicitly the separation between these two notions of solution for non-convex Hamiltonians.…”

Algorithm for overcoming the curse of dimensionality for state-dependent Hamilton-Jacobi equations

“…For more exposition of the minimax solution and development of this relationship with control problem, we refer the readers to [61] A more general notion of minimax viscosity solution (with a general non-convex but smooth Hamiltonion) which helps to recast the solution using a formulation that involves patching the graph of multivalued geometric solutions in a correct manner, e.g. in [5,4]. In the convex case, under further assumption, the solution can be formulated as a mini-max saddle point problem of a functional over the spaces of curves [4].…”

“…in [5,4]. In the convex case, under further assumption, the solution can be formulated as a mini-max saddle point problem of a functional over the spaces of curves [4]. However, it is known that the general formulation of minimax solution coincide with the viscosity solution only if the dynamically programming principle (i.e.…”

“…With these assumptions at hand, we have the following lemmas. We would like to remark that the following is a more discrete version of the minimax formula appeared in [4].…”