Book contents
- Frontmatter
- Contents
- Denjoy subsystems and horseshoes
- Impact Hamiltonian systems and polygonal billiards
- Some remarks on the classical KAM theorem, following Pöschel
- Some recent developments in Arnold diffusion
- Viscosity solutions of the Hamilton–Jacobi equation on a noncompact manifold
- Holonomy and vortex structures in quantum hydrodynamics
- Surfaces of locally minimal flux
- A symplectic approach to Arnold diffusion problems
- Hamiltonian ODE, homogenization, and symplectic topology
- References
Viscosity solutions of the Hamilton–Jacobi equation on a noncompact manifold
Published online by Cambridge University Press: 10 May 2024
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Book contents
- Frontmatter
- Contents
- Denjoy subsystems and horseshoes
- Impact Hamiltonian systems and polygonal billiards
- Some remarks on the classical KAM theorem, following Pöschel
- Some recent developments in Arnold diffusion
- Viscosity solutions of the Hamilton–Jacobi equation on a noncompact manifold
- Holonomy and vortex structures in quantum hydrodynamics
- Surfaces of locally minimal flux
- A symplectic approach to Arnold diffusion problems
- Hamiltonian ODE, homogenization, and symplectic topology
- References
Summary
We study the continuous viscosity solutions of the evolutionary Hamilton–Jacobi equation ∂t U (t, x) + H (x , ∂x U (x, t)) = 0 on [0, + ∞[xM, where H is a Tonelli Hamiltonian on the noncompact manifold M. We establish that all such solutions are given by the Lax–Oleinik formula. Moreover, we show that a finite everywhere Lax–Oleinik evolution is necessarily continuous and a viscosity solution on [0, + ∞ [xM. The goal is also to provide a convenient reference for the evolutionary Hamilton–Jacobi equation for Tonelli Hamiltonians on noncompact manifolds.
Keywords
- Type
- Chapter
- Information
- Hamiltonian SystemsDynamics, Analysis, Applications, pp. 111 - 172Publisher: Cambridge University PressPrint publication year: 2024
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