A Representation Formula of the Viscosity Solution of the Contact Hamilton-Jacobi Equation and Its Applications

Abstract

Assume that M is a closed, connected and smooth Riemannian manifold. The authors consider the evolutionary Hamilton-Jacobi equation

\{\begin{array}{ll}{\mathrm{\partial }}_{t}u(x,t)+H(x,u(x,t),{\mathrm{\partial }}_{x}u(x,t))=0,& (x,t)\in M\times (0,+\mathrm{\infty }),\\ u(x,0)=\phi (x),& \end{array}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{cases}\partial_{t}u(x,t)+H(x,u(x,t),\partial_{x}u(x,t))=0, & (x,t) \in M \times (0, +\infty), \\u(x,0)=\varphi(x), &\end{cases}$$\end{document}

where φ ∈ C(M) and the stationary one

H(x,u(x),{\mathrm{\partial }}_{x}u(x))=0,

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$H(x,u(x),\partial_{x}u(x))=0,$$\end{document}

where H(x,u,p) is continuous, convex and coercive in p, uniformly Lipschitz in u. By introducing a solution semigroup, the authors provide a representation formula of the viscosity solution of the evolutionary equation. As its applications, they obtain a necessary and sufficient condition for the existence of the viscosity solutions of the stationary equations. Moreover, they prove a new comparison theorem depending on the neighborhood of the projected Aubry set essentially, which is different from the one for the Hamilton-Jacobi equation independent of u.

Keywords

Weak KAM theory / Hamilton-Jacobi equations / Aubry sets

References

[1] BalesGAn Introduction to the Theory of Viscosity Solutions for First-Order Hamilton-Jacobi Equations and Applications, 2013, Heidelberg, Springer-Verlag: 49-109

[2] BallJ M, MizelV J. One-dimensional variational problems whose minimizers do not satisfy the Euler-Lagrange equation. Arch. Rational Mech. Anal., 1985, 90: 325-388

[3] BettiolP, MaricondaC. A new variational inequality in the calculus of variations and Lipschitz regularity of minimizers. J. Differential Equations, 2020, 5(268): 2332-2367

[4] ButtazzoG, GiaquintaM, HildebrandtSOne-Dimensional Variational Problems, an Introduction, 1998, New York, Clarendon

[5] CannarsaP, SinestrariCSemiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control, 2004, New York, Springer-Verlag 58

[6] CannarsaP, ChengW, WangK, YanJ. Herglotz’ generalized variational principle and contact type Hamilton-Jacobi equations. Trends in Control Theory and Partial Differential Equations, 2019, Cham, Springer-Verlag: 39-67 32

[7] CannarsaP, ChengW, JinL, et al.. Herglotz’ variational principle and Lax-Oleinik evolution. J. Math. Pures Appl., 2020, 141: 99-136

[8] ChenQ, ChengW, IshiiH, ZhaoK. Vanishing contact structure problem and convergence of the viscosity solutions. Comm. Partial Differential Equations, 2019, 44(9): 801-836

[9] CrandallM, EvansL, LionsP-L. Some properties of viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc., 1984, 282(2): 487-502

[10] CrandallM, IshiiH, LionsP-L. User’s guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.), 1992, 27(1): 1-67

[11] CrandallM, LionsP-L. Viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc., 1983, 277: 1-42

[12] DaviniA, SiconolfiA. A generalized dynamical approach to the large time behavior of solutions of Hamilton-Jacobi equations. SIAM J. Math. Anal., 2006, 38(2): 478-502

[13] FathiAWeak KAM Theorems in Lagrangian Dynamics, 2008, Cambridge, Cambridge University Press10th preliminary version

[14] FathiA, SiconolfiA. PDE aspects of Aubry-Mather theory for quasiconvex Hamiltonians. Calc. Var. Partial Differential Equations, 2005, 2: 185-228

[15] HerglotzGBerührungstransformationen, 1930

[16] HongJ, ChengW, HuS, ZhaoK. Representation formulas for contact type Hamilton-Jacobi equations. J. Dynam. Differential Equations, 2022, 34(3): 2315-2327

[17] IchiharaN, IshiiH. Long-time behavior of solutions of Hamilton-Jacobi equations with convex and coercive Hamiltonians. Arch. Rational Mech. Anal., 2009, 194: 383-419

[18] IshiiH. Perron’s methods for Hamilton-Jacobi equations. Duke Math. J., 1987, 55: 369-384

[19] IshiiH, WangK, WangL, YanJ. Hamilton-Jacobi equations with their Hamiltonians depending Lipschitz continuously on the unknown. Comm. Partial Differential Equations, 2022, 47: 417-452

[20] JinL, WangL, YanJ. A representation formula of viscosity solutions to weakly coupled systems of Hamilton-Jacobi equations with applications to regularizing effect. J. Differential Equations, 2020, 268: 2012-2039

[21] LiuQ, TorresP, WangC. Contact Hamiltonian dynamics: Variational principles, invariants, completeness and periodic behavior. Annals of Physics, 2018, 395: 26-44

[22] MasoG DAn Introduction to Γ-convergence, 1993, Boston, Birkhäuser 8

[23] WangK, WangL, YanJ. Implict variational principle for contact Hamiltonian systems. Nonlinearity, 2017, 30: 492-515

[24] WangK, WangL, YanJ. Variational principle for contact Hamiltonian systems and its applications. J. Math. Pures Appl., 2019, 123: 167-200

[25] WangK, WangL, YanJ. Aubry-Mather theory for contact Hamiltonian systems. Commun. Math. Phys., 2019, 366: 981-1023

[26] WangK, WangL, YanJ. Weak KAM solutions of Hamilton-Jacobi equations with decreasing dependence on unknown functions. J. Differential Equations, 2021, 286: 411-432

[27] WangK, WangL, YanJ. Aubry-Mather theory for contact Hamiltonian systems II. Discrete Contin. Dyn. Syst., 2022, 42: 555-595

RIGHTS & PERMISSIONS

The Editorial Office of CAM and Springer-Verlag Berlin Heidelberg