Viscosity solution of Hamilton-Jacobi equation by a limiting minmax method

TL;DR

This paper demonstrates that for non-convex Hamiltonians, iterating the minimax procedure over decreasing time intervals converges to the viscosity solution of the Hamilton-Jacobi equation, linking geometric and viscosity solutions.

Contribution

It establishes a method to recover the viscosity solution from the minimax solution through iterative short-time procedures for non-convex Hamiltonians.

Findings

Iterating minimax over shorter intervals converges to viscosity solution.

Viscosity and minimax solutions do not generally coincide for non-convex Hamiltonians.

The method provides a bridge between geometric and viscosity solutions.

Abstract

For non convex Hamiltonians, the viscosity solution and the more geometric minimax solution of the Hamilton-Jacobi equation do not coincide in general. They are nevertheless related: we show that iterating the minimax procedure during shorter and shorter time intervals one recovers the viscosity solution.