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doi: 10.3934/dcds.2021079

A multiplicity result for orthogonal geodesic chords in Finsler disks

Università di Camerino, Scuola di Scienze e Tecnologie, Camerino (MC), Italy

* Corresponding author: Dario Corona

Received  August 2020 Revised  March 2021 Published  May 2021

In this paper, we study the existence and multiplicity problems for orthogonal Finsler geodesic chords in a manifold with boundary which is homeomorphic to a $ N $-dimensional disk. Under a suitable assumption, which is weaker than convexity, we prove that, if the Finsler metric is reversible, then there are at least $ N $ orthogonal Finsler geodesic chords that are geometrically distinct. If the reversibility assumption does not hold, then there are at least two orthogonal Finsler geodesic chords with different values of the energy functional.

Citation: Dario Corona. A multiplicity result for orthogonal geodesic chords in Finsler disks. Discrete & Continuous Dynamical Systems, doi: 10.3934/dcds.2021079
References:
[1]

A. Abbondandolo and A. Figalli, High action orbits for Tonelli Langrangians and superlinear Hamiltonians on compact configuration spaces, J. Differential Equations, 234 (2007), 626-653.  doi: 10.1016/j.jde.2006.10.015.  Google Scholar

[2]

L. Asselle, On the existence of Euler-Lagrange orbits satisfying the conormal boundary conditions, J. Funct. Anal., 271 (2016), 3513-3553.  doi: 10.1016/j.jfa.2016.08.023.  Google Scholar

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R. BartoloE. CaponioA. V. Germinario and M. Sánchez, Convex domains of Finsler and Riemannian manifolds, Calc. Var. Partial Differential Equations, 40 (2011), 335-356.  doi: 10.1007/s00526-010-0343-1.  Google Scholar

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A. Canino, Periodic solutions of Lagrangian systems on manifolds with boundary, Nonlinear Anal., 16 (1991), 567-586.  doi: 10.1016/0362-546X(91)90029-Z.  Google Scholar

[5]

E. CaponioM. Á. Javaloyes and A. Masiello, On the energy functional on Finsler manifolds and applications to stationary spacetimes, Math. Ann., 351 (2011), 365-392.  doi: 10.1007/s00208-010-0602-7.  Google Scholar

[6]

D. Corona, A multiplicity result for Euler-Lagrange orbits satisfying the conormal boundary conditions, J. Fixed Point Theory Appl., 22 (2020), 60, 32 pp. doi: 10.1007/s11784-020-00795-4.  Google Scholar

[7]

D. Corona and F. Giannoni, Brake orbits for Hamiltonian systems of classical type via Finsler geodesics, (to appear) Google Scholar

[8]

R. Giambò, F. Giannoni and P. Piccione, Orthogonal geodesic chords, brake orbits and homoclinic orbits in Riemannian manifolds, Adv. Differential Equations, 10 (2005), 931-960. arXiv: math/0410391  Google Scholar

[9]

R. GiambòF. Giannoni and P. Piccione, On the multiplicity of orthogonal geodesies in Riemannian manifold with concave boundary. Applications to brake orbits and homoclinics, Adv. Nonlinear Stud., 9 (2009), 763-782.  doi: 10.1515/ans-2009-0409.  Google Scholar

[10]

R. GiambòF. Giannoni and P. Piccione, Existence of orthogonal geodesic chords on Riemannian manifolds with concave boundary and Homeomorphic to the $N$-dimensional disk, Nonlinear Anal., 73 (2010), 290-337.  doi: 10.1016/j.na.2010.03.019.  Google Scholar

[11]

R. GiambòF. Giannoni and P. Piccione, Multiple brake orbits and Homoclinics in Riemannian manifolds, Arch. Ration. Mech. Anal., 200 (2011), 691-724.  doi: 10.1007/s00205-010-0371-1.  Google Scholar

[12]

R. GiambòF. Giannoni and P. Piccione, Multiple brake orbits in $m$-dimensional disks, Calc. Var. Partial Differential Equations, 54 (2015), 2553-2580.  doi: 10.1007/s00526-015-0875-5.  Google Scholar

[13]

R. Giambò, F. Giannoni and P. Piccione, Multiple orthogonal geodesic chords in nonconvex Riemannian disks using obstacles, Calc. Var. Partial Differential Equations, 57 (2018), Paper No. 117, 26 pp. doi: 10.1007/s00526-018-1394-y.  Google Scholar

[14]

R. Giambò, F. Giannoni and P. Piccione, Multiple orthogonal geodesic chords and a proof of Seifert's conjecture on brake orbits, arXiv: 2002.09687 Google Scholar

[15]

F. Giannoni and P. Majer, On the effect of the domain on the number of orthogonal geodesic chords, Differential Geom. Appl., 7 (1997), 341-364.  doi: 10.1016/S0926-2245(96)00055-1.  Google Scholar

[16]

F. Giannoni and A. Masiello, On the existence of geodesics on stationary Lorentz manifolds with convex boundary, J. Funct. Anal., 101 (1991), 340-369.  doi: 10.1016/0022-1236(91)90162-X.  Google Scholar

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D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.  Google Scholar

[18]

W. B. Gordon, The existence of geodesics joining two given points, J. Differential Geometry, 9 (1974), 443-450.  doi: 10.4310/jdg/1214432420.  Google Scholar

[19]

C. Liu and D. Zhang, Seifert conjecture in the even convex case, Comm. Pure Appl. Math., 67 (2014), 1563-1604.  doi: 10.1002/cpa.21525.  Google Scholar

[20]

A. Marino and D. Scolozzi, Geodetiche con ostacolo, Boll. UMI B (6), 2 (1983), 1-31.   Google Scholar

[21]

F. Mercuri, The critical points theory for the closed geodesics problem, Math. Z., 156 (1977), 231-245.  doi: 10.1007/BF01214411.  Google Scholar

[22]

D. Scolozzi, A result of local uniqueness for geodesics on manifolds with boundary, Boll. Un. Mat. Ital., 5-B (1986), 309–327. doi: 11587/118622.  Google Scholar

[23]

H. Seifert, Periodische Bewegungen mechanischer Systeme, Math. Z., 51 (1948), 197-216.  doi: 10.1007/BF01291002.  Google Scholar

[24]

Z. Shen, Lectures on Finsler Geometry, World Scientific Publishing, 2001. doi: 10.1142/9789812811622.  Google Scholar

[25]

A. Weinstein, Periodic orbits for convex Hamiltonian systems, Ann. of Math., 108 (1978), 507-518.  doi: 10.2307/1971185.  Google Scholar

show all references

References:
[1]

A. Abbondandolo and A. Figalli, High action orbits for Tonelli Langrangians and superlinear Hamiltonians on compact configuration spaces, J. Differential Equations, 234 (2007), 626-653.  doi: 10.1016/j.jde.2006.10.015.  Google Scholar

[2]

L. Asselle, On the existence of Euler-Lagrange orbits satisfying the conormal boundary conditions, J. Funct. Anal., 271 (2016), 3513-3553.  doi: 10.1016/j.jfa.2016.08.023.  Google Scholar

[3]

R. BartoloE. CaponioA. V. Germinario and M. Sánchez, Convex domains of Finsler and Riemannian manifolds, Calc. Var. Partial Differential Equations, 40 (2011), 335-356.  doi: 10.1007/s00526-010-0343-1.  Google Scholar

[4]

A. Canino, Periodic solutions of Lagrangian systems on manifolds with boundary, Nonlinear Anal., 16 (1991), 567-586.  doi: 10.1016/0362-546X(91)90029-Z.  Google Scholar

[5]

E. CaponioM. Á. Javaloyes and A. Masiello, On the energy functional on Finsler manifolds and applications to stationary spacetimes, Math. Ann., 351 (2011), 365-392.  doi: 10.1007/s00208-010-0602-7.  Google Scholar

[6]

D. Corona, A multiplicity result for Euler-Lagrange orbits satisfying the conormal boundary conditions, J. Fixed Point Theory Appl., 22 (2020), 60, 32 pp. doi: 10.1007/s11784-020-00795-4.  Google Scholar

[7]

D. Corona and F. Giannoni, Brake orbits for Hamiltonian systems of classical type via Finsler geodesics, (to appear) Google Scholar

[8]

R. Giambò, F. Giannoni and P. Piccione, Orthogonal geodesic chords, brake orbits and homoclinic orbits in Riemannian manifolds, Adv. Differential Equations, 10 (2005), 931-960. arXiv: math/0410391  Google Scholar

[9]

R. GiambòF. Giannoni and P. Piccione, On the multiplicity of orthogonal geodesies in Riemannian manifold with concave boundary. Applications to brake orbits and homoclinics, Adv. Nonlinear Stud., 9 (2009), 763-782.  doi: 10.1515/ans-2009-0409.  Google Scholar

[10]

R. GiambòF. Giannoni and P. Piccione, Existence of orthogonal geodesic chords on Riemannian manifolds with concave boundary and Homeomorphic to the $N$-dimensional disk, Nonlinear Anal., 73 (2010), 290-337.  doi: 10.1016/j.na.2010.03.019.  Google Scholar

[11]

R. GiambòF. Giannoni and P. Piccione, Multiple brake orbits and Homoclinics in Riemannian manifolds, Arch. Ration. Mech. Anal., 200 (2011), 691-724.  doi: 10.1007/s00205-010-0371-1.  Google Scholar

[12]

R. GiambòF. Giannoni and P. Piccione, Multiple brake orbits in $m$-dimensional disks, Calc. Var. Partial Differential Equations, 54 (2015), 2553-2580.  doi: 10.1007/s00526-015-0875-5.  Google Scholar

[13]

R. Giambò, F. Giannoni and P. Piccione, Multiple orthogonal geodesic chords in nonconvex Riemannian disks using obstacles, Calc. Var. Partial Differential Equations, 57 (2018), Paper No. 117, 26 pp. doi: 10.1007/s00526-018-1394-y.  Google Scholar

[14]

R. Giambò, F. Giannoni and P. Piccione, Multiple orthogonal geodesic chords and a proof of Seifert's conjecture on brake orbits, arXiv: 2002.09687 Google Scholar

[15]

F. Giannoni and P. Majer, On the effect of the domain on the number of orthogonal geodesic chords, Differential Geom. Appl., 7 (1997), 341-364.  doi: 10.1016/S0926-2245(96)00055-1.  Google Scholar

[16]

F. Giannoni and A. Masiello, On the existence of geodesics on stationary Lorentz manifolds with convex boundary, J. Funct. Anal., 101 (1991), 340-369.  doi: 10.1016/0022-1236(91)90162-X.  Google Scholar

[17]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.  Google Scholar

[18]

W. B. Gordon, The existence of geodesics joining two given points, J. Differential Geometry, 9 (1974), 443-450.  doi: 10.4310/jdg/1214432420.  Google Scholar

[19]

C. Liu and D. Zhang, Seifert conjecture in the even convex case, Comm. Pure Appl. Math., 67 (2014), 1563-1604.  doi: 10.1002/cpa.21525.  Google Scholar

[20]

A. Marino and D. Scolozzi, Geodetiche con ostacolo, Boll. UMI B (6), 2 (1983), 1-31.   Google Scholar

[21]

F. Mercuri, The critical points theory for the closed geodesics problem, Math. Z., 156 (1977), 231-245.  doi: 10.1007/BF01214411.  Google Scholar

[22]

D. Scolozzi, A result of local uniqueness for geodesics on manifolds with boundary, Boll. Un. Mat. Ital., 5-B (1986), 309–327. doi: 11587/118622.  Google Scholar

[23]

H. Seifert, Periodische Bewegungen mechanischer Systeme, Math. Z., 51 (1948), 197-216.  doi: 10.1007/BF01291002.  Google Scholar

[24]

Z. Shen, Lectures on Finsler Geometry, World Scientific Publishing, 2001. doi: 10.1142/9789812811622.  Google Scholar

[25]

A. Weinstein, Periodic orbits for convex Hamiltonian systems, Ann. of Math., 108 (1978), 507-518.  doi: 10.2307/1971185.  Google Scholar

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