A Smooth, Recurrent, Non-Periodic Viscosity Solution of the Hamilton-Jacobi Equation

TL;DR

This paper introduces a novel method to construct smooth, recurrent, non-periodic viscosity solutions for the Hamilton-Jacobi equation on compact manifolds, expanding understanding beyond traditional convergence limitations.

Contribution

The authors develop a new approach to generate smooth viscosity solutions that are recurrent and non-periodic, and analyze the non-wandering set of the Lax-Oleinik operator.

Findings

Constructed smooth, recurrent, non-periodic solutions on manifolds of dimension ≥2.

Described the non-wandering set of the Lax-Oleinik operator.

Analyzed the action of the operator on omega-limit sets.

Abstract

Viscosity solutions of the Hamilton-Jacobi equation were introduced by Lions and Crandall. For Tonelli Hamiltonians, these solutions are generated by the Lax-Oleinik operator. It is known that this operator converges in the autonomous framework, but this convergence fails in the general cases. In this paper, we introduce a method to construct smooth, recurrent, non-periodic viscosity solutions on fixed compact manifolds $M$ of dimension 2 or higher. Additionally, we provide a detailed description of the non-wandering set of the Lax-Oleinik operator and identify its action on various omega-limit sets.