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November  2016, 36(11): 6487-6522. doi: 10.3934/dcds.2016080

Weak KAM theory for HAMILTON-JACOBI equations depending on unknown functions

Xifeng Su 1, , Lin Wang 2, and  Jun Yan 3,

1. 

School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China
2. 

Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084, China
3. 

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

Received  September 2015 Revised  July 2016 Published  August 2016

We consider the evolutionary Hamilton-Jacobi equation depending on the unknown function with the continuous initial condition on a connected closed manifold. Under certain assumptions on $H(x,u,p)$ with respect to $u$ and $p$, we provide an implicit variational principle. By introducing an implicitly defined solution semigroup and an admissible value set $\mathcal{C}_H$, we extend weak KAM theory to certain more general cases, in which $H$ depends on the unknown function $u$ explicitly. As an application, we show that for $0\notin \mathcal{C}_H$, as $t\rightarrow +\infty$, the viscosity solution of \begin{equation*} \begin{cases} \partial_tu(x,t)+H(x,u(x,t),\partial_xu(x,t))=0,\\ u(x,0)=\varphi(x), \end{cases} \end{equation*} diverges, otherwise for $0\in \mathcal{C}_H$, it converges to a weak KAM solution of the stationary Hamilton-Jacobi equation \begin{equation*} H(x,u(x),\partial_xu(x))=0. \end{equation*}
Keywords: Hamilton-Jacobi equations, viscosity solutions., Weak KAM theory.
Mathematics Subject Classification: Primary: 37J50, 35F21; Secondary: 35D4.
Citation: Xifeng Su, Lin Wang, Jun Yan. Weak KAM theory for HAMILTON-JACOBI equations depending on unknown functions. Discrete & Continuous Dynamical Systems, 2016, 36 (11) : 6487-6522. doi: 10.3934/dcds.2016080
References:
[1]

V. I. Arnold, Geometric Methods in The Theory of Ordinary Differential Equations, Springer-Verlag, New York, 1983.  Google Scholar

[2]

V. I. Arnold, Lectures on Partial Differential Equations, Springer, Berlin Heidelberg, 2004. doi: 10.1007/978-3-662-05441-3.  Google Scholar

[3]

S. Aubry, The twist map, the extended Frenkel-Kontorova model and the devil's staircase, Phys. D., 7 (1983), 240-258. doi: 10.1016/0167-2789(83)90129-X.  Google Scholar

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S. Aubry and P. Y. Le Daeron, The discrete Frenkel-Kontorova model and its extensions I: Exact results for the ground states, Phys. D., 8 (1983), 381-422. doi: 10.1016/0167-2789(83)90233-6.  Google Scholar

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G. Buttazzo, M. Giaquinta and S. Hildebrandt, One-dimensional Variational Problems, An Introduction, (Oxford Lecture Series in Mathematics and Its Applications), Clarendon Press Oxford, 1998.  Google Scholar

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P. Cannarsa and C. Sinestrari, Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control, Vol. 58, Springer, 2004.  Google Scholar

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M. G. Crandall, L. C. Evans and P.-L. Lions, Some properties of viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 282 (1984), 487-502. doi: 10.1090/S0002-9947-1984-0732102-X.  Google Scholar

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M. G. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1-67. doi: 10.1090/S0273-0979-1992-00266-5.  Google Scholar

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G. Contreras, R. Iturriaga, G. P. Paternain and M. Paternain, Lagrangian graphs, minimizing measures and Mañé's critical values, Geom. Funct. Anal., 8 (1998), 788-809. doi: 10.1007/s000390050074.  Google Scholar

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M. G. Crandall and P.-L. 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

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[13]

A. Davini and A. Siconolfi, A generalized dynamical approach to the large time behavior of solutions of Hamilton-Jacobi equations, SIAM J. Math. Anal., 38 (2006), 478-502. doi: 10.1137/050621955.  Google Scholar

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A. Douglis, Solutions in the large for multi-dimensional, non-linear partial differential equations of first order, Ann. Inst. Fourier (Grenoble), 15 (1965), 1-35. doi: 10.5802/aif.208.  Google Scholar

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W. E, Aubry-Mather theory and periodic solutions of the forced Burgers equation, Comm. Pure Appl. Math., 52 (1999), 811-828. doi: 10.1002/(SICI)1097-0312(199907)52:7<811::AID-CPA2>3.0.CO;2-D.  Google Scholar

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[17]

A. Fathi, Sur la convergence du semi-groupe de Lax-Oleinik, C. R. Acad. Sci. Paris Sér. I Math., 327 (1998), 267-270. doi: 10.1016/S0764-4442(98)80144-4.  Google Scholar

[18]

A. Fathi, Weak KAM Theorem in Lagrangian Dynamics, Preliminary Version Number 10, 2008. Google Scholar

[19]

A. Fathi and J. N. 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

[20]

N. Ichihara and H. Ishii, Long-time behavior of solutions of Hamilton-Jacobi equations with convex and coercive Hamiltonians, Arch. Ration. Mech. Anal., 194 (2009), 383-419. doi: 10.1007/s00205-008-0170-0.  Google Scholar

[21]

J. N. Mather, Existence of quasi periodic orbits for twist homeomorphisms of the annulus, Topology, 21 (1982), 457-467. doi: 10.1016/0040-9383(82)90023-4.  Google Scholar

[22]

J. N. Mather, More Denjoy minimal sets for area preserving diffeomorphisms, Comment. Math. Helv., 60 (1985), 508-557. doi: 10.1007/BF02567431.  Google Scholar

[23]

J. N. Mather, A criterion for the non-existence of invariant circle, Publ. Math. IHES, 63 (1986), 301-309. doi: 10.1007/BF02831625.  Google Scholar

[24]

J. N. Mather, Modulus of continuity for Peierls's barrier, Periodic Solutions of Hamiltonian Systems and Related Topics, ed. P.H.Rabinowitz et al. NATO ASI Series C 209, Reidel: Dordrecht, (1987), 177-202.  Google Scholar

[25]

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

[26]

G. Namah and J.-M. Roquejoffre, Remarks on the long time behavior of the solutions of Hamilton-Jacobi equations, Commun. Partial Differ. Equ., 24 (1999), 883-893. doi: 10.1080/03605309908821451.  Google Scholar

[27]

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

show all references

References:
[1]

V. I. Arnold, Geometric Methods in The Theory of Ordinary Differential Equations, Springer-Verlag, New York, 1983.  Google Scholar

[2]

V. I. Arnold, Lectures on Partial Differential Equations, Springer, Berlin Heidelberg, 2004. doi: 10.1007/978-3-662-05441-3.  Google Scholar

[3]

S. Aubry, The twist map, the extended Frenkel-Kontorova model and the devil's staircase, Phys. D., 7 (1983), 240-258. doi: 10.1016/0167-2789(83)90129-X.  Google Scholar

[4]

S. Aubry and P. Y. Le Daeron, The discrete Frenkel-Kontorova model and its extensions I: Exact results for the ground states, Phys. D., 8 (1983), 381-422. doi: 10.1016/0167-2789(83)90233-6.  Google Scholar

[5]

G. Barles, Solutions de Viscosité des Équations de Hamilton-Jacobi Mathématiques & Applications, (Berlin) 17, Springer, Paris, 1994.  Google Scholar

[6]

G. Buttazzo, M. Giaquinta and S. Hildebrandt, One-dimensional Variational Problems, An Introduction, (Oxford Lecture Series in Mathematics and Its Applications), Clarendon Press Oxford, 1998.  Google Scholar

[7]

P. Cannarsa and C. Sinestrari, Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control, Vol. 58, Springer, 2004.  Google Scholar

[8]

M. G. Crandall, L. C. Evans and P.-L. Lions, Some properties of viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 282 (1984), 487-502. doi: 10.1090/S0002-9947-1984-0732102-X.  Google Scholar

[9]

M. G. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1-67. doi: 10.1090/S0273-0979-1992-00266-5.  Google Scholar

[10]

G. Contreras, R. Iturriaga, G. P. Paternain and M. Paternain, Lagrangian graphs, minimizing measures and Mañé's critical values, Geom. Funct. Anal., 8 (1998), 788-809. doi: 10.1007/s000390050074.  Google Scholar

[11]

M. G. Crandall and P.-L. 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

[12]

A. Davini, A. Fathi, R. Iturriaga and M. Zavidovique., Convergence of the solutions of the discounted Hamilton-Jacobi equation, Invent. Math., 105 (2016), 1-27. doi: 10.1007/s00222-016-0648-6.  Google Scholar

[13]

A. Davini and A. Siconolfi, A generalized dynamical approach to the large time behavior of solutions of Hamilton-Jacobi equations, SIAM J. Math. Anal., 38 (2006), 478-502. doi: 10.1137/050621955.  Google Scholar

[14]

A. Douglis, Solutions in the large for multi-dimensional, non-linear partial differential equations of first order, Ann. Inst. Fourier (Grenoble), 15 (1965), 1-35. doi: 10.5802/aif.208.  Google Scholar

[15]

W. E, Aubry-Mather theory and periodic solutions of the forced Burgers equation, Comm. Pure Appl. Math., 52 (1999), 811-828. doi: 10.1002/(SICI)1097-0312(199907)52:7<811::AID-CPA2>3.0.CO;2-D.  Google Scholar

[16]

A. Fathi, Théorème KAM faible et théorie de Mather sur les systèmes lagrangiens, C. R. Acad. Sci. Paris Sér. I Math., 324 (1997), 1043-1046. doi: 10.1016/S0764-4442(97)87883-4.  Google Scholar

[17]

A. Fathi, Sur la convergence du semi-groupe de Lax-Oleinik, C. R. Acad. Sci. Paris Sér. I Math., 327 (1998), 267-270. doi: 10.1016/S0764-4442(98)80144-4.  Google Scholar

[18]

A. Fathi, Weak KAM Theorem in Lagrangian Dynamics, Preliminary Version Number 10, 2008. Google Scholar

[19]

A. Fathi and J. N. 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

[20]

N. Ichihara and H. Ishii, Long-time behavior of solutions of Hamilton-Jacobi equations with convex and coercive Hamiltonians, Arch. Ration. Mech. Anal., 194 (2009), 383-419. doi: 10.1007/s00205-008-0170-0.  Google Scholar

[21]

J. N. Mather, Existence of quasi periodic orbits for twist homeomorphisms of the annulus, Topology, 21 (1982), 457-467. doi: 10.1016/0040-9383(82)90023-4.  Google Scholar

[22]

J. N. Mather, More Denjoy minimal sets for area preserving diffeomorphisms, Comment. Math. Helv., 60 (1985), 508-557. doi: 10.1007/BF02567431.  Google Scholar

[23]

J. N. Mather, A criterion for the non-existence of invariant circle, Publ. Math. IHES, 63 (1986), 301-309. doi: 10.1007/BF02831625.  Google Scholar

[24]

J. N. Mather, Modulus of continuity for Peierls's barrier, Periodic Solutions of Hamiltonian Systems and Related Topics, ed. P.H.Rabinowitz et al. NATO ASI Series C 209, Reidel: Dordrecht, (1987), 177-202.  Google Scholar

[25]

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

[26]

G. Namah and J.-M. Roquejoffre, Remarks on the long time behavior of the solutions of Hamilton-Jacobi equations, Commun. Partial Differ. Equ., 24 (1999), 883-893. doi: 10.1080/03605309908821451.  Google Scholar

[27]

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

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