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arXiv · 2505.08700

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

Skander Charfi
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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.

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Skander Charfi. 2025-05-13. A Smooth, Recurrent, Non-Periodic Viscosity Solution of the Hamilton-Jacobi Equation. https://arxiv.org/abs/2505.08700

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