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Viscosity solutions of the Hamilton–Jacobi equation on a noncompact manifold

Published online by Cambridge University Press:  10 May 2024

By
Albert Fathi
Edited by
Albert Fathi ,
Philip J. Morrison ,
Tere M-Seara  and
Sergei Tabachnikov
Albert Fathi
Affiliation:
Georgia Institute of Technology
Philip J. Morrison
Affiliation:
University of Texas, Austin
Tere M-Seara
Affiliation:
Universitat Politècnica de Catalunya, Barcelona
Sergei Tabachnikov
Affiliation:
Pennsylvania State University
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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

Hamilton–Jacobi equation
Type
Chapter
Information
Hamiltonian Systems
Dynamics, Analysis, Applications
 , pp. 111 - 172
Publisher: Cambridge University Press
Print publication year: 2024

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References

Bardi, M. and Capuzzo-Dolcetta, I., Optimal control and viscosity solutions of Hamilton– Jacobi–Bellman equations, Birkhäuser, Boston, 1997.CrossRefGoogle Scholar
Barles, G., Solutions de viscosité des équations de Hamilton–Jacobi, Springer, Paris, 1994.Google Scholar
Buttazzo, G., Giaquinta, M., and Hildebrandt, S., One-dimensional variational problems, Oxford Lecture Series in Mathematics and its Applications 15, Oxford Univ. Press, New York, 1998.Google Scholar
Cannarsa, P. and Sinestrari, C., Semiconcave functions, Hamilton–Jacobi equations, and optimal control, Progress in Nonlinear Differential Equations and their Applications 58, Birkhäuser, Boston, 2004.Google Scholar
Clarke, F. H., Methods of dynamic and nonsmooth optimization, CBMS-NSF Regional Conference Series in Applied Mathematics 57, Soc. Industr. Appl. Math., Philadelphia, 1989.CrossRefGoogle Scholar
Fathi, A., “Weak KAM theorem in Lagrangian dynamics”, book draft version 10, 2005, available at https://www.math.u-bordeaux.fr/~pthieull/Recherche/KamFaible/Publications/ Fathi2008_01.pdf.Google Scholar
Fathi, A., “Weak KAM from a PDE point of view: viscosity solutions of the Hamilton–Jacobi equation and Aubry set”, Proc. Roy. Soc. Edinburgh Sect. A 120 (2012), 1193–1236.CrossRefGoogle Scholar
Fathi, A. and Figalli, A., “Optimal transportation on non-compact manifolds”, Israel J. Math. 175 (2010), 1–59.CrossRefGoogle Scholar
Fathi, A. and Maderna, E., “Weak KAM theorem on non compact manifolds”, Nonlinear Diff. Eq. Appl. 14 (2007), 1–27.CrossRefGoogle Scholar
Ishii, H., “A short introduction to viscosity solutions and the large time behavior of solutions of Hamilton–Jacobi equations”, in Hamilton–Jacobi equations: approximations, edited by Achdou, Y. et al., Lecture Notes in Mathematics 2074, Springer, 2013.Google Scholar
Liu, Q., Wang, K., Wang, L., and Yan, J., “A necessary and sufficient condition for convergence of the Lax–Oleinik semigroup for reversible Hamiltonians on ℝn”, J. Diff. Eq. 261 (2016), 5289–5305.CrossRefGoogle Scholar
Zavidovique, M., “Strict sub-solutions and Mañé potential in discrete weak KAM theory”, Comment. Math. Helv. 87 (2012), 1–39.CrossRefGoogle Scholar

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