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March  2016, 36(3): 1649-1659. doi: 10.3934/dcds.2016.36.1649

Lipschitz dependence of viscosity solutions of Hamilton-Jacobi equations with respect to the parameter

Kaizhi Wang 1, and  Jun Yan 2,

1. 

Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China
2. 

School of Mathematical Sciences, Fudan University, Shanghai Key Laboratory for Contemporary Applied Mathematics, Shanghai 200433, China

Received  October 2014 Revised  March 2015 Published  August 2015

Let $M$ be a closed and smooth manifold and $H_\varepsilon:T^*M\to\mathbf{R}^1$ be a family of Tonelli Hamiltonians for $\varepsilon\geq0$ small. For each $\varphi\in C(M,\mathbf{R}^1)$, $T^\varepsilon_t\varphi(x)$ is the unique viscosity solution of the Cauchy problem \begin{align*} \left\{ \begin{array}{ll} d_tw+H_\varepsilon(x,d_xw)=0, & \ \mathrm{in}\ M\times(0,+\infty),\\ w|_{t=0}=\varphi, & \ \mathrm{on}\ M, \end{array} \right. \end{align*} where $T^\varepsilon_t$ is the Lax-Oleinik operator associated with $H_\varepsilon$. A result of Fathi asserts that the uniform limit, for $t\to+\infty$, of $T^\varepsilon_t\varphi+c_\varepsilon t$ exists and the limit $\bar{\varphi}_\varepsilon$ is a viscosity solution of the stationary Hamilton-Jacobi equation \begin{align*} H_\varepsilon(x,d_xu)=c_\varepsilon, \end{align*} where $c_\varepsilon$ is the unique $k$ for which the equation $H_\varepsilon(x,d_xu)=k$ admits viscosity solutions. In the present paper we discuss the continuous dependence of the viscosity solution $\bar{\varphi}_\varepsilon$ with respect to the parameter $\varepsilon$.
Keywords: weak KAM theory., Lipschitz dependence, Hamilton-Jacobi equations, viscosity solutions, large-time behavior.
Mathematics Subject Classification: Primary: 37J50; Secondary: 70H2.
Citation: Kaizhi Wang, Jun Yan. Lipschitz dependence of viscosity solutions of Hamilton-Jacobi equations with respect to the parameter. Discrete & Continuous Dynamical Systems, 2016, 36 (3) : 1649-1659. doi: 10.3934/dcds.2016.36.1649
References:
[1]

P. Bernard, Smooth critical sub-solutions of the Hamilton-Jacobi equation, Math. Res. Lett., 14 (2007), 503-511. doi: 10.4310/MRL.2007.v14.n3.a14.  Google Scholar

[2]

C. Cheng and J. Yan, Existence of diffusion orbits in a priori unstable Hamiltonian systems, J. Differential Geom., 67 (2004), 457-517.  Google Scholar

[3]

C. Cheng and J. Yan, Arnold diffusion in Hamiltonian systems: A priori unstable case, J. Differential Geom., 82 (2009), 229-277.  Google Scholar

[4]

G. Contreras, Action potential and weak KAM solutions, Calc. Var. Partial Differential Equations, 13 (2001), 427-458. doi: 10.1007/s005260100081.  Google Scholar

[5]

M. Crandall and P. Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 277 (1983), 1-42. doi: 10.1090/S0002-9947-1983-0690039-8.  Google Scholar

[6]

A. Fathi and J. Mather, Failure of convergence of the Lax-Oleinik semi-group in the time-periodic case, Bull. Soc. Math. France, 128 (2000), 473-483.  Google Scholar

[7]

A. Fathi, Weak KAM Theorems in Lagrangian Dynamics, Seventh preliminary version, Pisa, 2005. Google Scholar

[8]

M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, 583, Springer Verlag, Berlin, New York, 1977.  Google Scholar

[9]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511809187.  Google Scholar

[10]

R. Mañé, Lagrangian flows: The dynamics of globally minimizing orbits, Bol. Soc. Brasil. Mat. (N.S.), 28 (1997), 141-153. doi: 10.1007/BF01233389.  Google Scholar

[11]

J. Mather, Action minimizing invariant measures for positive definite Lagrangian systems, Math. Z., 207 (1991), 169-207. doi: 10.1007/BF02571383.  Google Scholar

[12]

J. Mather, Variational construction of connecting orbits, Ann. Inst. Fourier (Grenoble), 43 (1993), 1349-1386. doi: 10.5802/aif.1377.  Google Scholar

[13]

J. Palis and W. de Melo, Geometric Theory of Dynamical Systems. An Introduction, Springer-Verlag, New York, 1982.  Google Scholar

[14]

A. Sorrentino, Lecture Notes on Mather's Theory for Lagrangian Systems, preprint, 2010. Google Scholar

[15]

K. Wang and J. Yan, A new kind of Lax-Oleinik type operator with parameters for time-periodic positive definite Lagrangian systems, Commun. Math. Phys., 309 (2012), 663-691. doi: 10.1007/s00220-011-1375-x.  Google Scholar

[16]

A. Weinstein, Symplectic manifolds and their Lagrangian submanifolds, Advances in Math., 6 (1971), 329-346. doi: 10.1016/0001-8708(71)90020-X.  Google Scholar

show all references

References:
[1]

P. Bernard, Smooth critical sub-solutions of the Hamilton-Jacobi equation, Math. Res. Lett., 14 (2007), 503-511. doi: 10.4310/MRL.2007.v14.n3.a14.  Google Scholar

[2]

C. Cheng and J. Yan, Existence of diffusion orbits in a priori unstable Hamiltonian systems, J. Differential Geom., 67 (2004), 457-517.  Google Scholar

[3]

C. Cheng and J. Yan, Arnold diffusion in Hamiltonian systems: A priori unstable case, J. Differential Geom., 82 (2009), 229-277.  Google Scholar

[4]

G. Contreras, Action potential and weak KAM solutions, Calc. Var. Partial Differential Equations, 13 (2001), 427-458. doi: 10.1007/s005260100081.  Google Scholar

[5]

M. Crandall and P. Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 277 (1983), 1-42. doi: 10.1090/S0002-9947-1983-0690039-8.  Google Scholar

[6]

A. Fathi and J. Mather, Failure of convergence of the Lax-Oleinik semi-group in the time-periodic case, Bull. Soc. Math. France, 128 (2000), 473-483.  Google Scholar

[7]

A. Fathi, Weak KAM Theorems in Lagrangian Dynamics, Seventh preliminary version, Pisa, 2005. Google Scholar

[8]

M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, 583, Springer Verlag, Berlin, New York, 1977.  Google Scholar

[9]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511809187.  Google Scholar

[10]

R. Mañé, Lagrangian flows: The dynamics of globally minimizing orbits, Bol. Soc. Brasil. Mat. (N.S.), 28 (1997), 141-153. doi: 10.1007/BF01233389.  Google Scholar

[11]

J. Mather, Action minimizing invariant measures for positive definite Lagrangian systems, Math. Z., 207 (1991), 169-207. doi: 10.1007/BF02571383.  Google Scholar

[12]

J. Mather, Variational construction of connecting orbits, Ann. Inst. Fourier (Grenoble), 43 (1993), 1349-1386. doi: 10.5802/aif.1377.  Google Scholar

[13]

J. Palis and W. de Melo, Geometric Theory of Dynamical Systems. An Introduction, Springer-Verlag, New York, 1982.  Google Scholar

[14]

A. Sorrentino, Lecture Notes on Mather's Theory for Lagrangian Systems, preprint, 2010. Google Scholar

[15]

K. Wang and J. Yan, A new kind of Lax-Oleinik type operator with parameters for time-periodic positive definite Lagrangian systems, Commun. Math. Phys., 309 (2012), 663-691. doi: 10.1007/s00220-011-1375-x.  Google Scholar

[16]

A. Weinstein, Symplectic manifolds and their Lagrangian submanifolds, Advances in Math., 6 (1971), 329-346. doi: 10.1016/0001-8708(71)90020-X.  Google Scholar

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