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

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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

Abstract
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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 and 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.
[2]	C. Cheng and J. Yan, Existence of diffusion orbits in a priori unstable Hamiltonian systems, J. Differential Geom., 67 (2004), 457-517.
[3]	C. Cheng and J. Yan, Arnold diffusion in Hamiltonian systems: A priori unstable case, J. Differential Geom., 82 (2009), 229-277.
[4]	G. Contreras, Action potential and weak KAM solutions, Calc. Var. Partial Differential Equations, 13 (2001), 427-458. doi: 10.1007/s005260100081.
[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.
[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.
[7]	A. Fathi, Weak KAM Theorems in Lagrangian Dynamics, Seventh preliminary version, Pisa, 2005.
[8]	M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, 583, Springer Verlag, Berlin, New York, 1977.
[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.
[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.
[11]	J. Mather, Action minimizing invariant measures for positive definite Lagrangian systems, Math. Z., 207 (1991), 169-207. doi: 10.1007/BF02571383.
[12]	J. Mather, Variational construction of connecting orbits, Ann. Inst. Fourier (Grenoble), 43 (1993), 1349-1386. doi: 10.5802/aif.1377.
[13]	J. Palis and W. de Melo, Geometric Theory of Dynamical Systems. An Introduction, Springer-Verlag, New York, 1982.
[14]	A. Sorrentino, Lecture Notes on Mather's Theory for Lagrangian Systems, preprint, 2010.
[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.
[16]	A. Weinstein, Symplectic manifolds and their Lagrangian submanifolds, Advances in Math., 6 (1971), 329-346. doi: 10.1016/0001-8708(71)90020-X.

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.
[2]	C. Cheng and J. Yan, Existence of diffusion orbits in a priori unstable Hamiltonian systems, J. Differential Geom., 67 (2004), 457-517.
[3]	C. Cheng and J. Yan, Arnold diffusion in Hamiltonian systems: A priori unstable case, J. Differential Geom., 82 (2009), 229-277.
[4]	G. Contreras, Action potential and weak KAM solutions, Calc. Var. Partial Differential Equations, 13 (2001), 427-458. doi: 10.1007/s005260100081.
[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.
[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.
[7]	A. Fathi, Weak KAM Theorems in Lagrangian Dynamics, Seventh preliminary version, Pisa, 2005.
[8]	M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, 583, Springer Verlag, Berlin, New York, 1977.
[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.
[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.
[11]	J. Mather, Action minimizing invariant measures for positive definite Lagrangian systems, Math. Z., 207 (1991), 169-207. doi: 10.1007/BF02571383.
[12]	J. Mather, Variational construction of connecting orbits, Ann. Inst. Fourier (Grenoble), 43 (1993), 1349-1386. doi: 10.5802/aif.1377.
[13]	J. Palis and W. de Melo, Geometric Theory of Dynamical Systems. An Introduction, Springer-Verlag, New York, 1982.
[14]	A. Sorrentino, Lecture Notes on Mather's Theory for Lagrangian Systems, preprint, 2010.
[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.
[16]	A. Weinstein, Symplectic manifolds and their Lagrangian submanifolds, Advances in Math., 6 (1971), 329-346. doi: 10.1016/0001-8708(71)90020-X.

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