• Previous Article
    Asymptotic and quenching behaviors of semilinear parabolic systems with singular nonlinearities
  • CPAA Home
  • This Issue
  • Next Article
    Global regularity estimates for Neumann problems of elliptic operators with coefficients having a BMO anti-symmetric part in NTA domains
doi: 10.3934/cpaa.2022009
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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

*Corresponding author

Received  February 2021 Revised  October 2021 Early access December 2021

Fund Project: Partially supported by the NNSF 11271044 of 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.

Citation: Kun Shi, Guangcun Lu. Higher P-symmetric Ekeland-Hofer capacities. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2022009
References:
[1]

A. Akopyan and R. Karasev, Estimating symplectic capacities from lengths of closed curves on the unit spheres, preprint, arXiv: 1801.00242. Google Scholar

[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.  Google Scholar

[3]

L. BaraccoM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

I. Ekeland and H. Hofer, Symplectic topology and Hamiltonian dynamics, Math. Z., 200 (1989), 355-378.  doi: 10.1007/BF01215653.  Google Scholar

[7]

I. Ekeland and H. Hofer, Symplectic topology and Haniltonian dynamics II, Math. Z., 203 (1990), 553-567.  doi: 10.1007/BF02570756.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

Rongrong Jin and Guangcun Lu, Generalizations of Ekeland-Hofer and Hofer-Zehnder symplectic capacities and applications, preprint, arXiv: 1903.01116v2. Google Scholar

[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.  Google Scholar

[13]

R. R. Jin and G. C. Lu, Coisotropic Ekeland-Hofer capacities, preprint, arXiv: 1910.14474. Google Scholar

[14]

V. G. B. Ramos, Symplectic embeddings and the Lagrangian bidisk, Duke Math. J., 166 (2017), 1703-1738.  doi: 10.1215/00127094-0000011X.  Google Scholar

[15]

J. C. Sikorav, Systémes Hamiltoniens et Topologie Symplectique, Ph.D thesis, Dipartimento di Matematica dell'Universitá di Pisa, 1990. Google Scholar

[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.  Google Scholar

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. Google Scholar

[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.  Google Scholar

[3]

L. BaraccoM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

I. Ekeland and H. Hofer, Symplectic topology and Hamiltonian dynamics, Math. Z., 200 (1989), 355-378.  doi: 10.1007/BF01215653.  Google Scholar

[7]

I. Ekeland and H. Hofer, Symplectic topology and Haniltonian dynamics II, Math. Z., 203 (1990), 553-567.  doi: 10.1007/BF02570756.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

Rongrong Jin and Guangcun Lu, Generalizations of Ekeland-Hofer and Hofer-Zehnder symplectic capacities and applications, preprint, arXiv: 1903.01116v2. Google Scholar

[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.  Google Scholar

[13]

R. R. Jin and G. C. Lu, Coisotropic Ekeland-Hofer capacities, preprint, arXiv: 1910.14474. Google Scholar

[14]

V. G. B. Ramos, Symplectic embeddings and the Lagrangian bidisk, Duke Math. J., 166 (2017), 1703-1738.  doi: 10.1215/00127094-0000011X.  Google Scholar

[15]

J. C. Sikorav, Systémes Hamiltoniens et Topologie Symplectique, Ph.D thesis, Dipartimento di Matematica dell'Universitá di Pisa, 1990. Google Scholar

[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.  Google Scholar

[1]

Hui Liu, Duanzhi Zhang. Stable P-symmetric closed characteristics on partially symmetric compact convex hypersurfaces. Discrete & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & Continuous Dynamical Systems, 2021  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 & 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

Article outline

[Back to Top]