September  2016, 36(9): 5131-5162. doi: 10.3934/dcds.2016023

Periodic solutions of the planar N-center problem with topological constraints

1. 

Department of Mathematics, University of Toronto, 40 St. George St., Room 6290, Toronto, Ontario, M5S 2E4, Canada

Received  July 2015 Revised  November 2015 Published  May 2016

In the planar $N$-center problem, given a non-trivial free homotopy class of the configuration space satisfying certain conditions, we show that there is at least one collision-free $T$-periodic solution for any positive $T.$ The direct method of calculus of variations is used and the main difficulty is to show that minimizers under certain topological constraints are free of collision.
Citation: Guowei Yu. Periodic solutions of the planar N-center problem with topological constraints. Discrete & Continuous Dynamical Systems, 2016, 36 (9) : 5131-5162. doi: 10.3934/dcds.2016023
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show all references

References:
[1]

Progress in Nonlinear Differential Equations and their Applications, 10, Birkhäuser Boston, Inc., Boston, MA, 1993. doi: 10.1007/978-1-4612-0319-3.  Google Scholar

[2]

J. Funct. Anal., 82 (1989), 412-428. doi: 10.1016/0022-1236(89)90078-5.  Google Scholar

[3]

Arch. Ration. Mech. Anal., 190 (2008), 189-226. doi: 10.1007/s00205-008-0131-7.  Google Scholar

[4]

Arch. Ration. Mech. Anal., 207 (2013), 583-609. doi: 10.1007/s00205-012-0565-9.  Google Scholar

[5]

Calc. Var. Partial Differential Equations, 49 (2014), 391-429. doi: 10.1007/s00526-012-0587-z.  Google Scholar

[6]

Arch. Ration. Mech. Anal., 170 (2003), 247-276. doi: 10.1007/s00205-003-0277-2.  Google Scholar

[7]

Ann. of Math. (2), 167 (2008), 325-348. doi: 10.4007/annals.2008.167.325.  Google Scholar

[8]

in Proceedings of the International Congress of Mathematicians, Higher Ed. Press, Beijing, (2002), 279-294.  Google Scholar

[9]

in Geometry, mechanics, and dynamics, (2002), 287-308. doi: 10.1007/0-387-21791-6_9.  Google Scholar

[10]

Ann. of Math. (2), 152 (2000), 881-901. doi: 10.2307/2661357.  Google Scholar

[11]

Celestial Mech. Dynam. Astronom., 77 (2000), 139-152. doi: 10.1023/A:1008381001328.  Google Scholar

[12]

Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 1998.  Google Scholar

[13]

Invent. Math., 155 (2004), 305-362. doi: 10.1007/s00222-003-0322-7.  Google Scholar

[14]

Invent. Math., 185 (2011), 283-332. doi: 10.1007/s00222-010-0306-3.  Google Scholar

[15]

Trans. Amer. Math. Soc., 204 (1975), 113-135. doi: 10.1090/S0002-9947-1975-0377983-1.  Google Scholar

[16]

Amer. J. Math., 99 (1977), 961-971. doi: 10.2307/2373993.  Google Scholar

[17]

Israel J. Math., 51 (1985), 90-120. doi: 10.1007/BF02772960.  Google Scholar

[18]

Texts in Mathematics, 33. Springer-Verlag, New York, 1994.  Google Scholar

[19]

Nonlinearity, 11 (1998), 363-376. doi: 10.1088/0951-7715/11/2/011.  Google Scholar

[20]

Arch. Ration. Mech. Anal., 214 (2014), 77-98. doi: 10.1007/s00205-014-0753-x.  Google Scholar

[21]

NoDEA Nonlinear Differential Equations Appl., 21 (2014), 371-413. doi: 10.1007/s00030-013-0251-0.  Google Scholar

[22]

Discrete Contin. Dyn. Syst. - A, 32 (2012), 3245-3301. doi: 10.3934/dcds.2012.32.3245.  Google Scholar

[23]

Arch. Ration. Mech. Anal., 184 (2007), 465-493. doi: 10.1007/s00205-006-0030-8.  Google Scholar

[24]

Ph.D thesis, Universit Denis Diderot in Paris, 2002. Google Scholar

[25]

arXiv:1509.04999, 2015. Google Scholar

[26]

arXiv:0707.0078, 2015. Google Scholar

[27]

G. Yu, Periodic solutions of the rotating N-center and restricted N + 1-body problems with topological constraints,, work in progress., ().   Google Scholar

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