-
Previous Article
Existence of solutions for a class of quasilinear Schrödinger equation with a Kirchhoff-type
- CPAA Home
- This Issue
-
Next Article
Monotonicity and symmetry of positive solutions to degenerate quasilinear elliptic systems in half-spaces and strips
Higher P-symmetric Ekeland-Hofer capacities
Laboratory of Mathematics and Complex Systems (Ministry of Education), School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China |
This paper is devoted to the construction of analogues of higher Ekeland-Hofer symplectic capacities for $ P $-symmetric subsets in the standard symplectic space $ (\mathbb{R}^{2n},\omega_0) $, which is motivated by Long and Dong's study about $ P $-symmetric closed characteristics on $ P $-symmetric convex bodies. We study the relationship between these capacities and other capacities, and give some computation examples. Moreover, we also define higher real symmetric Ekeland-Hofer capacities as a complement of Jin and the second named author's recent study of the real symmetric analogue about the first Ekeland-Hofer capacity.
References:
| [1] |
A. Akopyan and R. Karasev, Estimating symplectic capacities from lengths of closed curves on the unit spheres, preprint, arXiv: 1801.00242. |
| [2] |
S. Artstein-Avidan and Y. Ostrover,
Bounds for Minkowski billiard trajectories in convex bodies, Int. Math. Res. Not., 2014 (2014), 165-193.
doi: 10.1093/imrn/rns216. |
| [3] |
L. Baracco, M. Fassina and S. Pinton,
On the Ekeland-Hofer symplectic capacities of the real bidisc, Pacific J. Math., 305 (2020), 423-446.
doi: 10.2140/pjm.2020.305.423. |
| [4] |
V. Benci,
On the critical point theory for indefinite functionals in the presence of symmetries, Trans. Amer. Math. Soc., 274 (1982), 533-572.
doi: 10.2307/1999120. |
| [5] |
Y. Dong and Y. Long,
Closed characteristics on partially symmetric compact convex hypersurfaces in $\mathbb{R}^{2n}$, J. Differ. Equ., 196 (2004), 226-248.
doi: 10.1016/S0022-0396(03)00168-2. |
| [6] |
I. Ekeland and H. Hofer,
Symplectic topology and Hamiltonian dynamics, Math. Z., 200 (1989), 355-378.
doi: 10.1007/BF01215653. |
| [7] |
I. Ekeland and H. Hofer,
Symplectic topology and Haniltonian dynamics II, Math. Z., 203 (1990), 553-567.
doi: 10.1007/BF02570756. |
| [8] |
E. R. Fadell and P. H. Rabinowitz,
Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems, Invent. Math., 45 (1978), 139-174.
doi: 10.1007/BF01390270. |
| [9] |
H. Hofer and E. Zehnder,
Periodic solutions on hypersurfaces and a result by C. Viterbo, Invent. Math., 90 (1987), 1-9.
doi: 10.1007/BF01389030. |
| [10] |
H, Hofer and E. Zehnder, Symplectic Invariants and Hamiltonian Dynamics, 1$^{st}$ edition, Birkhüser Verlag, Basel, 1994.
doi: 10.1007/978-3-0348-8540-9. |
| [11] |
Rongrong Jin and Guangcun Lu, Generalizations of Ekeland-Hofer and Hofer-Zehnder symplectic capacities and applications, preprint, arXiv: 1903.01116v2. |
| [12] |
R. R. Jin and G. C. Lu,
Representation formula for symmetrical symplectic capacity and applications, Discrete Contin. Dyn. Syst., 40 (2020), 4705-4765.
doi: 10.3934/dcds.2020199. |
| [13] |
R. R. Jin and G. C. Lu, Coisotropic Ekeland-Hofer capacities, preprint, arXiv: 1910.14474. |
| [14] |
V. G. B. Ramos,
Symplectic embeddings and the Lagrangian bidisk, Duke Math. J., 166 (2017), 1703-1738.
doi: 10.1215/00127094-0000011X. |
| [15] |
J. C. Sikorav, Systémes Hamiltoniens et Topologie Symplectique, Ph.D thesis, Dipartimento di Matematica dell'Universitá di Pisa, 1990. |
| [16] |
A. Szulkin,
An index theory and existence of multiple brake orbits for star-shaped Hamiltonian systems, Math. Ann., 283 (1989), 241-255.
doi: 10.1007/BF01446433. |
show all references
References:
| [1] |
A. Akopyan and R. Karasev, Estimating symplectic capacities from lengths of closed curves on the unit spheres, preprint, arXiv: 1801.00242. |
| [2] |
S. Artstein-Avidan and Y. Ostrover,
Bounds for Minkowski billiard trajectories in convex bodies, Int. Math. Res. Not., 2014 (2014), 165-193.
doi: 10.1093/imrn/rns216. |
| [3] |
L. Baracco, M. Fassina and S. Pinton,
On the Ekeland-Hofer symplectic capacities of the real bidisc, Pacific J. Math., 305 (2020), 423-446.
doi: 10.2140/pjm.2020.305.423. |
| [4] |
V. Benci,
On the critical point theory for indefinite functionals in the presence of symmetries, Trans. Amer. Math. Soc., 274 (1982), 533-572.
doi: 10.2307/1999120. |
| [5] |
Y. Dong and Y. Long,
Closed characteristics on partially symmetric compact convex hypersurfaces in $\mathbb{R}^{2n}$, J. Differ. Equ., 196 (2004), 226-248.
doi: 10.1016/S0022-0396(03)00168-2. |
| [6] |
I. Ekeland and H. Hofer,
Symplectic topology and Hamiltonian dynamics, Math. Z., 200 (1989), 355-378.
doi: 10.1007/BF01215653. |
| [7] |
I. Ekeland and H. Hofer,
Symplectic topology and Haniltonian dynamics II, Math. Z., 203 (1990), 553-567.
doi: 10.1007/BF02570756. |
| [8] |
E. R. Fadell and P. H. Rabinowitz,
Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems, Invent. Math., 45 (1978), 139-174.
doi: 10.1007/BF01390270. |
| [9] |
H. Hofer and E. Zehnder,
Periodic solutions on hypersurfaces and a result by C. Viterbo, Invent. Math., 90 (1987), 1-9.
doi: 10.1007/BF01389030. |
| [10] |
H, Hofer and E. Zehnder, Symplectic Invariants and Hamiltonian Dynamics, 1$^{st}$ edition, Birkhüser Verlag, Basel, 1994.
doi: 10.1007/978-3-0348-8540-9. |
| [11] |
Rongrong Jin and Guangcun Lu, Generalizations of Ekeland-Hofer and Hofer-Zehnder symplectic capacities and applications, preprint, arXiv: 1903.01116v2. |
| [12] |
R. R. Jin and G. C. Lu,
Representation formula for symmetrical symplectic capacity and applications, Discrete Contin. Dyn. Syst., 40 (2020), 4705-4765.
doi: 10.3934/dcds.2020199. |
| [13] |
R. R. Jin and G. C. Lu, Coisotropic Ekeland-Hofer capacities, preprint, arXiv: 1910.14474. |
| [14] |
V. G. B. Ramos,
Symplectic embeddings and the Lagrangian bidisk, Duke Math. J., 166 (2017), 1703-1738.
doi: 10.1215/00127094-0000011X. |
| [15] |
J. C. Sikorav, Systémes Hamiltoniens et Topologie Symplectique, Ph.D thesis, Dipartimento di Matematica dell'Universitá di Pisa, 1990. |
| [16] |
A. Szulkin,
An index theory and existence of multiple brake orbits for star-shaped Hamiltonian systems, Math. Ann., 283 (1989), 241-255.
doi: 10.1007/BF01446433. |
| [1] |
Hui Liu, Duanzhi Zhang. Stable P-symmetric closed characteristics on partially symmetric compact convex hypersurfaces. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 877-893. doi: 10.3934/dcds.2016.36.877 |
| [2] |
Duanzhi Zhang. $P$-cyclic symmetric closed characteristics on compact convex $P$-cyclic symmetric hypersurface in R2n. Discrete and Continuous Dynamical Systems, 2013, 33 (2) : 947-964. doi: 10.3934/dcds.2013.33.947 |
| [3] |
Lei Liu, Li Wu. Multiplicity of closed characteristics on $ P $-symmetric compact convex hypersurfaces in $ \mathbb{R}^{2n} $. Discrete and Continuous Dynamical Systems, 2021, 41 (6) : 2635-3652. doi: 10.3934/dcds.2020378 |
| [4] |
L. F. Cheung, C. K. Law, M. C. Leung. On a class of rotationally symmetric $p$-harmonic maps. Communications on Pure and Applied Analysis, 2017, 16 (6) : 1941-1955. doi: 10.3934/cpaa.2017095 |
| [5] |
Jian Lu, Huaiyu Jian. Topological degree method for the rotationally symmetric $L_p$-Minkowski problem. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 971-980. doi: 10.3934/dcds.2016.36.971 |
| [6] |
Wei Wang. Closed trajectories on symmetric convex Hamiltonian energy surfaces. Discrete and Continuous Dynamical Systems, 2012, 32 (2) : 679-701. doi: 10.3934/dcds.2012.32.679 |
| [7] |
Li-Xia Liu, Sanyang Liu, Chun-Feng Wang. Smoothing Newton methods for symmetric cone linear complementarity problem with the Cartesian $P$/$P_0$-property. Journal of Industrial and Management Optimization, 2011, 7 (1) : 53-66. doi: 10.3934/jimo.2011.7.53 |
| [8] |
Jan Burczak, P. Kaplický. Evolutionary, symmetric $p$-Laplacian. Interior regularity of time derivatives and its consequences. Communications on Pure and Applied Analysis, 2016, 15 (6) : 2401-2445. doi: 10.3934/cpaa.2016042 |
| [9] |
Lixing Han. An unconstrained optimization approach for finding real eigenvalues of even order symmetric tensors. Numerical Algebra, Control and Optimization, 2013, 3 (3) : 583-599. doi: 10.3934/naco.2013.3.583 |
| [10] |
François Lalonde, Yasha Savelyev. On the injectivity radius in Hofer's geometry. Electronic Research Announcements, 2014, 21: 177-185. doi: 10.3934/era.2014.21.177 |
| [11] |
Charles Curry, Stephen Marsland, Robert I McLachlan. Principal symmetric space analysis. Journal of Computational Dynamics, 2019, 6 (2) : 251-276. doi: 10.3934/jcd.2019013 |
| [12] |
Juraj Földes, Peter Poláčik. On asymptotically symmetric parabolic equations. Networks and Heterogeneous Media, 2012, 7 (4) : 673-689. doi: 10.3934/nhm.2012.7.673 |
| [13] |
Martin Kassabov. Symmetric groups and expanders. Electronic Research Announcements, 2005, 11: 47-56. |
| [14] |
Muhammad Hamid, Wei Wang. A symmetric property in the enhanced common index jump theorem with applications to the closed geodesic problem. Discrete and Continuous Dynamical Systems, 2022, 42 (4) : 1933-1948. doi: 10.3934/dcds.2021178 |
| [15] |
Jiao Du, Longjiang Qu, Chao Li, Xin Liao. Constructing 1-resilient rotation symmetric functions over $ {\mathbb F}_{p} $ with $ {q} $ variables through special orthogonal arrays. Advances in Mathematics of Communications, 2020, 14 (2) : 247-263. doi: 10.3934/amc.2020018 |
| [16] |
Barry Simon. Equilibrium measures and capacities in spectral theory. Inverse Problems and Imaging, 2007, 1 (4) : 713-772. doi: 10.3934/ipi.2007.1.713 |
| [17] |
Goro Akagi, Jun Kobayashi, Mitsuharu Ôtani. Principle of symmetric criticality and evolution equations. Conference Publications, 2003, 2003 (Special) : 1-10. doi: 10.3934/proc.2003.2003.1 |
| [18] |
Julián López-Gómez. Uniqueness of radially symmetric large solutions. Conference Publications, 2007, 2007 (Special) : 677-686. doi: 10.3934/proc.2007.2007.677 |
| [19] |
Ivica Martinjak, Mario-Osvin Pavčević. Symmetric designs possessing tactical decompositions. Advances in Mathematics of Communications, 2011, 5 (2) : 199-208. doi: 10.3934/amc.2011.5.199 |
| [20] |
Krzysztof A. Krakowski, Luís Machado, Fátima Silva Leite. A unifying approach for rolling symmetric spaces. Journal of Geometric Mechanics, 2021, 13 (1) : 145-166. doi: 10.3934/jgm.2020016 |
2020 Impact Factor: 1.916
Tools
Metrics
Other articles
by authors
[Back to Top]