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

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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

Abstract
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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 and 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.
[2]	V. I. Arnold, Lectures on Partial Differential Equations, Springer, Berlin Heidelberg, 2004. doi: 10.1007/978-3-662-05441-3.
[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.
[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.
[5]	G. Barles, Solutions de Viscosité des Équations de Hamilton-Jacobi Mathématiques & Applications, (Berlin) 17, Springer, Paris, 1994.
[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.
[7]	P. Cannarsa and C. Sinestrari, Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control, Vol. 58, Springer, 2004.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[18]	A. Fathi, Weak KAM Theorem in Lagrangian Dynamics, Preliminary Version Number 10, 2008.
[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.
[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.
[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.
[22]	J. N. Mather, More Denjoy minimal sets for area preserving diffeomorphisms, Comment. Math. Helv., 60 (1985), 508-557. doi: 10.1007/BF02567431.
[23]	J. N. Mather, A criterion for the non-existence of invariant circle, Publ. Math. IHES, 63 (1986), 301-309. doi: 10.1007/BF02831625.
[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.
[25]	J. N. Mather, Action minimizing invariant measures for positive definite Lagrangian systems, Math. Z., 207 (1991), 169-207. doi: 10.1007/BF02571383.
[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.
[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.

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References:

[1]	V. I. Arnold, Geometric Methods in The Theory of Ordinary Differential Equations, Springer-Verlag, New York, 1983.
[2]	V. I. Arnold, Lectures on Partial Differential Equations, Springer, Berlin Heidelberg, 2004. doi: 10.1007/978-3-662-05441-3.
[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.
[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.
[5]	G. Barles, Solutions de Viscosité des Équations de Hamilton-Jacobi Mathématiques & Applications, (Berlin) 17, Springer, Paris, 1994.
[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.
[7]	P. Cannarsa and C. Sinestrari, Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control, Vol. 58, Springer, 2004.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[18]	A. Fathi, Weak KAM Theorem in Lagrangian Dynamics, Preliminary Version Number 10, 2008.
[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.
[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.
[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.
[22]	J. N. Mather, More Denjoy minimal sets for area preserving diffeomorphisms, Comment. Math. Helv., 60 (1985), 508-557. doi: 10.1007/BF02567431.
[23]	J. N. Mather, A criterion for the non-existence of invariant circle, Publ. Math. IHES, 63 (1986), 301-309. doi: 10.1007/BF02831625.
[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.
[25]	J. N. Mather, Action minimizing invariant measures for positive definite Lagrangian systems, Math. Z., 207 (1991), 169-207. doi: 10.1007/BF02571383.
[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.
[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.

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