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Local singular characteristics on .

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Summary
This summary is machine-generated.

This study establishes a general uniqueness criterion for singular characteristics in Hamilton-Jacobi equations. It proves two notions of singular characteristics coincide for Tonelli Hamiltonians, resolving a key gap in the literature.

Keywords:
Hamilton–Jacobi equationSingular characteristicsViscosity solution

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Area of Science:

  • * Mathematical Physics
  • * Nonlinear Partial Differential Equations

Background:

  • * Viscosity solutions of Hamilton-Jacobi equations feature a singular set.
  • * This singular set propagates along singular characteristics defined by differential inclusions.
  • * Existing literature lacks a general uniqueness criterion for these singular characteristics.

Purpose of the Study:

  • * To establish a general uniqueness criterion for singular characteristics in Hamilton-Jacobi equations.
  • * To demonstrate the coincidence of two distinct notions of singular characteristics.
  • * To provide a uniqueness result for a specific class of singular characteristics.

Main Methods:

  • * Analysis of singular characteristics for Hamilton-Jacobi equations with Tonelli Hamiltonians.
  • * Proof of the coincidence of two different definitions of singular characteristics.
  • * Utilizing differential inclusions to characterize singular characteristic curves.

Main Results:

  • * Two distinct notions of singular characteristics are shown to coincide up to bi-Lipschitz reparameterization for Tonelli Hamiltonians.
  • * A significant consequence is a uniqueness result for singular characteristics defined by Khanin and Sobolevski.
  • * This work addresses the missing general uniqueness criterion for singular characteristics.

Conclusions:

  • * The paper provides a unified understanding of singular characteristics in Hamilton-Jacobi equations.
  • * The findings contribute to the theory of viscosity solutions and their singular sets.
  • * This research advances the study of dynamics for Lagrangian trajectories in this context.