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April  2022, 42(4): 1949-1970. doi: 10.3934/dcds.2021179

Global propagation of singularities for discounted Hamilton-Jacobi equations

1. 

School of Mathematical Sciences, Jiangsu University, Zhenjiang 212013, China

2. 

Department of Mathematics, Nanjing University, Nanjing 210093, China

3. 

School of Mathematical Sciences, Fudan University, Shanghai 200433, China

*Corresponding author: Kai Zhao

Received  June 2021 Revised  September 2021 Published  April 2022 Early access  November 2021

Fund Project: The first author is partly supported by National Natural Science Foundation of China (Grant No. 11801223, 11871267)

The main purpose of this paper is to study the global propagation of singularities of the viscosity solution to discounted Hamilton-Jacobi equation
$ \begin{align} \lambda v(x)+H( x, Dv(x) ) = 0 , \quad x\in \mathbb{R}^n. \quad\quad\quad (\mathrm{HJ}_{\lambda})\end{align} $
with fixed constant
$ \lambda\in \mathbb{R}^+ $
. We reduce the problem for equation
$(\mathrm{HJ}_{\lambda})$
into that for a time-dependent evolutionary Hamilton-Jacobi equation. We prove that the singularities of the viscosity solution of
$(\mathrm{HJ}_{\lambda})$
propagate along locally Lipschitz singular characteristics
$ {{\bf{x}}}(s):[0,t]\to \mathbb{R}^n $
and time
$ t $
can extend to
$ +\infty $
. Essentially, we use
$ \sigma $
-compactness of the Euclidean space which is different from the original construction in [4]. The local Lipschitz issue is a key technical difficulty to study the global result. As a application, we also obtain the homotopy equivalence between the singular locus of
$ u $
and the complement of Aubry set using the basic idea from [9].
Citation: Cui Chen, Jiahui Hong, Kai Zhao. Global propagation of singularities for discounted Hamilton-Jacobi equations. Discrete and Continuous Dynamical Systems, 2022, 42 (4) : 1949-1970. doi: 10.3934/dcds.2021179
References:
[1]

Y. Achdou, G. Barles, H. Ishii and G. L. Litvinov, Hamilton-Jacobi Equations: Approximations, Numerical Analysis and Applications, vol. 2074 of Lecture Notes in Mathematics, Springer, Heidelberg; Fondazione C.I.M.E., Florence, 2013, Lecture Notes from the CIME Summer School held in Cetraro, August 29–September 3, 2011, Edited by Paola Loreti and Nicoletta Anna Tchou, Fondazione CIME/CIME Foundation Subseries. doi: 10.1007/978-3-642-36433-4.

[2]

P. Albano and P. Cannarsa, Propagation of singularities for solutions of nonlinear first order partial differential equations, Arch. Ration. Mech. Anal., 162 (2002), 1-23.  doi: 10.1007/s002050100176.

[3]

A. Brown and C. Pearcy, An Introduction to Analysis, vol. 154, Graduate Texts in Mathematics, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-0787-0.

[4]

P. Cannarsa and W. Cheng, Generalized characteristics and Lax-Oleinik operators: Global theory, Calc. Var. Partial Differential Equations, 56 (2017), Art. 125, 31 pp. doi: 10.1007/s00526-017-1219-4.

[5] P. Cannarsa and W. Cheng, On and beyond propagation of singularities of viscosity solutions, Proceedings of the International Consortium of Chinese Mathematicians, Int. Press, Boston, MA, 2020. 
[6]

P. Cannarsa and W. Cheng, Local singular characteristics on $\mathbb{R}^2$, Boll. Unione Mat. Ital., 14 (2021), 483-504.  doi: 10.1007/s40574-021-00279-4.

[7]

P. Cannarsa and W. Cheng, Singularities of solutions of Hamilton-Jacobi equations, Milan Journal of Mathematics, 89 (2021), 187-215.  doi: 10.1007/s00032-021-00330-1.

[8]

P. CannarsaW. Cheng and A. Fathi, On the topology of the set of singularities of a solution to the Hamilton-Jacobi equation, C. R. Math. Acad. Sci. Paris, 355 (2017), 176-180.  doi: 10.1016/j.crma.2016.12.004.

[9]

P. Cannarsa, W. Cheng and A. Fathi, Singularities of solutions of time dependent Hamilton-Jacobi equations. Applications to Riemannian geometry, Publications Mathématiques de L'IHÉS, 133 (2021), 327–366. doi: 10.1007/s10240-021-00125-5.

[10]

P. CannarsaW. ChengL. JinK. Wang and J. Yan, Herglotz' variational principle and Lax-Oleinik evolution, J. Math. Pures Appl., 141 (2020), 99-136.  doi: 10.1016/j.matpur.2020.07.002.

[11]

P. CannarsaW. ChengM. Mazzola and K. Wang, Global generalized characteristics for the Dirichlet problem for Hamilton-Jacobi equations at a supercritical energy level, SIAM J. Math. Anal., 51 (2019), 4213-4244.  doi: 10.1137/18M1203547.

[12]

P. CannarsaM. Mazzola and C. Sinestrari, Global propagation of singularities for time dependent Hamilton-Jacobi equations, Discrete Contin. Dyn. Syst., 35 (2015), 4225-4239.  doi: 10.3934/dcds.2015.35.4225.

[13]

P. Cannarsa and C. Sinestrari, Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control, vol. 58 of Progress in Nonlinear Differential Equations and their Applications, Birkhäuser Boston, Inc., Boston, MA, 2004.

[14]

P. Cannarsa and Y. Yu, Singular dynamics for semiconcave functions, J. Eur. Math. Soc., 11 (2009), 999-1024.  doi: 10.4171/JEMS/173.

[15]

C. ChenW. Cheng and Q. Zhang, Lasry-Lions approximations for discounted Hamilton-Jacobi equations, J. Differential Equations, 265 (2018), 719-732.  doi: 10.1016/j.jde.2018.03.010.

[16]

W. Cheng and J. Hong, Local strict singular characteristics: Cauchy problem with smooth initial data, arXiv: 2103.06217.

[17]

F. H. Clarke and R. B. Vinter, Regularity properties of solutions to the basic problem in the calculus of variations, Trans. Amer. Math. Soc., 289 (1985), 73-98.  doi: 10.1090/S0002-9947-1985-0779053-3.

[18]

J. Dugundji, Topology, Allyn and Bacon, Inc., Boston, Mass., 1966.

[19]

A. Fathi, Solutions KAM faibles conjuguées et barrières de Peierls, C. R. Acad. Sci. Paris Sér. I Math., 325 (1997), 649–652. doi: 10.1016/S0764-4442(97)84777-5.

[20]

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.

[21]

A. Fathi and E. Maderna, Weak KAM theorem on non compact manifolds, NoDEA Nonlinear Differential Equations Appl., 14 (2007), 1-27.  doi: 10.1007/s00030-007-2047-6.

[22]

K. Khanin and A. Sobolevski, On dynamics of Lagrangian trajectories for Hamilton-Jacobi equations, Arch. Ration. Mech. Anal., 219 (2016), 861-885.  doi: 10.1007/s00205-015-0910-x.

[23]

S. Marò and A. Sorrentino, Aubry-Mather theory for conformally symplectic systems, Comm. Math. Phys., 354 (2017), 775-808.  doi: 10.1007/s00220-017-2900-3.

[24]

T. Strömberg, Propagation of singularities along broken characteristics, Nonlinear Anal., 85 (2013), 93-109.  doi: 10.1016/j.na.2013.02.024.

[25]

K. WangL. Wang and J. Yan, Aubry–mather theory for contact Hamiltonian systems, Comm. Math. Phys., 366 (2019), 981-1023.  doi: 10.1007/s00220-019-03362-2.

[26]

Y. Yu, A simple proof of the propagation of singularities for solutions of Hamilton-Jacobi equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 5 (2006), 439-444.  doi: 10.2422/2036-2145.2006.4.01.

show all references

References:
[1]

Y. Achdou, G. Barles, H. Ishii and G. L. Litvinov, Hamilton-Jacobi Equations: Approximations, Numerical Analysis and Applications, vol. 2074 of Lecture Notes in Mathematics, Springer, Heidelberg; Fondazione C.I.M.E., Florence, 2013, Lecture Notes from the CIME Summer School held in Cetraro, August 29–September 3, 2011, Edited by Paola Loreti and Nicoletta Anna Tchou, Fondazione CIME/CIME Foundation Subseries. doi: 10.1007/978-3-642-36433-4.

[2]

P. Albano and P. Cannarsa, Propagation of singularities for solutions of nonlinear first order partial differential equations, Arch. Ration. Mech. Anal., 162 (2002), 1-23.  doi: 10.1007/s002050100176.

[3]

A. Brown and C. Pearcy, An Introduction to Analysis, vol. 154, Graduate Texts in Mathematics, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-0787-0.

[4]

P. Cannarsa and W. Cheng, Generalized characteristics and Lax-Oleinik operators: Global theory, Calc. Var. Partial Differential Equations, 56 (2017), Art. 125, 31 pp. doi: 10.1007/s00526-017-1219-4.

[5] P. Cannarsa and W. Cheng, On and beyond propagation of singularities of viscosity solutions, Proceedings of the International Consortium of Chinese Mathematicians, Int. Press, Boston, MA, 2020. 
[6]

P. Cannarsa and W. Cheng, Local singular characteristics on $\mathbb{R}^2$, Boll. Unione Mat. Ital., 14 (2021), 483-504.  doi: 10.1007/s40574-021-00279-4.

[7]

P. Cannarsa and W. Cheng, Singularities of solutions of Hamilton-Jacobi equations, Milan Journal of Mathematics, 89 (2021), 187-215.  doi: 10.1007/s00032-021-00330-1.

[8]

P. CannarsaW. Cheng and A. Fathi, On the topology of the set of singularities of a solution to the Hamilton-Jacobi equation, C. R. Math. Acad. Sci. Paris, 355 (2017), 176-180.  doi: 10.1016/j.crma.2016.12.004.

[9]

P. Cannarsa, W. Cheng and A. Fathi, Singularities of solutions of time dependent Hamilton-Jacobi equations. Applications to Riemannian geometry, Publications Mathématiques de L'IHÉS, 133 (2021), 327–366. doi: 10.1007/s10240-021-00125-5.

[10]

P. CannarsaW. ChengL. JinK. Wang and J. Yan, Herglotz' variational principle and Lax-Oleinik evolution, J. Math. Pures Appl., 141 (2020), 99-136.  doi: 10.1016/j.matpur.2020.07.002.

[11]

P. CannarsaW. ChengM. Mazzola and K. Wang, Global generalized characteristics for the Dirichlet problem for Hamilton-Jacobi equations at a supercritical energy level, SIAM J. Math. Anal., 51 (2019), 4213-4244.  doi: 10.1137/18M1203547.

[12]

P. CannarsaM. Mazzola and C. Sinestrari, Global propagation of singularities for time dependent Hamilton-Jacobi equations, Discrete Contin. Dyn. Syst., 35 (2015), 4225-4239.  doi: 10.3934/dcds.2015.35.4225.

[13]

P. Cannarsa and C. Sinestrari, Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control, vol. 58 of Progress in Nonlinear Differential Equations and their Applications, Birkhäuser Boston, Inc., Boston, MA, 2004.

[14]

P. Cannarsa and Y. Yu, Singular dynamics for semiconcave functions, J. Eur. Math. Soc., 11 (2009), 999-1024.  doi: 10.4171/JEMS/173.

[15]

C. ChenW. Cheng and Q. Zhang, Lasry-Lions approximations for discounted Hamilton-Jacobi equations, J. Differential Equations, 265 (2018), 719-732.  doi: 10.1016/j.jde.2018.03.010.

[16]

W. Cheng and J. Hong, Local strict singular characteristics: Cauchy problem with smooth initial data, arXiv: 2103.06217.

[17]

F. H. Clarke and R. B. Vinter, Regularity properties of solutions to the basic problem in the calculus of variations, Trans. Amer. Math. Soc., 289 (1985), 73-98.  doi: 10.1090/S0002-9947-1985-0779053-3.

[18]

J. Dugundji, Topology, Allyn and Bacon, Inc., Boston, Mass., 1966.

[19]

A. Fathi, Solutions KAM faibles conjuguées et barrières de Peierls, C. R. Acad. Sci. Paris Sér. I Math., 325 (1997), 649–652. doi: 10.1016/S0764-4442(97)84777-5.

[20]

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.

[21]

A. Fathi and E. Maderna, Weak KAM theorem on non compact manifolds, NoDEA Nonlinear Differential Equations Appl., 14 (2007), 1-27.  doi: 10.1007/s00030-007-2047-6.

[22]

K. Khanin and A. Sobolevski, On dynamics of Lagrangian trajectories for Hamilton-Jacobi equations, Arch. Ration. Mech. Anal., 219 (2016), 861-885.  doi: 10.1007/s00205-015-0910-x.

[23]

S. Marò and A. Sorrentino, Aubry-Mather theory for conformally symplectic systems, Comm. Math. Phys., 354 (2017), 775-808.  doi: 10.1007/s00220-017-2900-3.

[24]

T. Strömberg, Propagation of singularities along broken characteristics, Nonlinear Anal., 85 (2013), 93-109.  doi: 10.1016/j.na.2013.02.024.

[25]

K. WangL. Wang and J. Yan, Aubry–mather theory for contact Hamiltonian systems, Comm. Math. Phys., 366 (2019), 981-1023.  doi: 10.1007/s00220-019-03362-2.

[26]

Y. Yu, A simple proof of the propagation of singularities for solutions of Hamilton-Jacobi equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 5 (2006), 439-444.  doi: 10.2422/2036-2145.2006.4.01.

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