---
OA_type: free access
_id: '17231'
abstract:
- lang: eng
  text: In the class of projective billiards, which contains the usual billiards,
    we exhibit counter-examples to Ivrii's conjecture, which states that in any planar
    billiard with smooth boundary the set of periodic orbits has zero measure. The
    counter-examples are polygons admitting a 2-parameters family of n-periodic orbits,
    with n being either 3 or any even integer greater than 4.
article_processing_charge: No
article_type: original
author:
- first_name: Corentin
  full_name: Fiorebe, Corentin
  id: 06619f18-9070-11eb-847d-d1ee780bd88b
  last_name: Fiorebe
citation:
  ama: Fiorebe C. Examples of projective billiards with open sets of periodic orbits.
    <i>Discrete and Continuous Dynamical Systems- Series A</i>. 2024;44(11):3287-3301.
    doi:<a href="https://doi.org/10.3934/dcds.2024059">10.3934/dcds.2024059</a>
  apa: Fiorebe, C. (2024). Examples of projective billiards with open sets of periodic
    orbits. <i>Discrete and Continuous Dynamical Systems- Series A</i>. American Institute
    of Mathematical Sciences. <a href="https://doi.org/10.3934/dcds.2024059">https://doi.org/10.3934/dcds.2024059</a>
  chicago: Fiorebe, Corentin. “Examples of Projective Billiards with Open Sets of
    Periodic Orbits.” <i>Discrete and Continuous Dynamical Systems- Series A</i>.
    American Institute of Mathematical Sciences, 2024. <a href="https://doi.org/10.3934/dcds.2024059">https://doi.org/10.3934/dcds.2024059</a>.
  ieee: C. Fiorebe, “Examples of projective billiards with open sets of periodic orbits,”
    <i>Discrete and Continuous Dynamical Systems- Series A</i>, vol. 44, no. 11. American
    Institute of Mathematical Sciences, pp. 3287–3301, 2024.
  ista: Fiorebe C. 2024. Examples of projective billiards with open sets of periodic
    orbits. Discrete and Continuous Dynamical Systems- Series A. 44(11), 3287–3301.
  mla: Fiorebe, Corentin. “Examples of Projective Billiards with Open Sets of Periodic
    Orbits.” <i>Discrete and Continuous Dynamical Systems- Series A</i>, vol. 44,
    no. 11, American Institute of Mathematical Sciences, 2024, pp. 3287–301, doi:<a
    href="https://doi.org/10.3934/dcds.2024059">10.3934/dcds.2024059</a>.
  short: C. Fiorebe, Discrete and Continuous Dynamical Systems- Series A 44 (2024)
    3287–3301.
corr_author: '1'
date_created: 2024-07-14T22:01:10Z
date_published: 2024-11-01T00:00:00Z
date_updated: 2026-06-18T17:54:16Z
day: '01'
ddc:
- '500'
department:
- _id: VaKa
doi: 10.3934/dcds.2024059
external_id:
  isi:
  - '001230091000001'
intvolume: '        44'
isi: 1
issue: '11'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.3934/dcds.2024059
month: '11'
oa: 1
oa_version: Published Version
page: 3287-3301
publication: Discrete and Continuous Dynamical Systems- Series A
publication_identifier:
  eissn:
  - 1553-5231
  issn:
  - 1078-0947
publication_status: published
publisher: American Institute of Mathematical Sciences
quality_controlled: '1'
scopus_import: '1'
status: public
title: Examples of projective billiards with open sets of periodic orbits
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 44
year: '2024'
...
---
OA_place: repository
OA_type: green
_id: '22030'
abstract:
- lang: eng
  text: We consider the mass-subcritical NLS in dimensions d>=3 with radial initial
    data. In the defocusing case, we prove that any solution that remains bounded
    in the critical Sobolev space throughout its lifespan must be global and scatter.
    In the focusing case, we prove the existence of a threshold solution that has
    a compact flow.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Rowan
  full_name: Killip, Rowan
  last_name: Killip
- first_name: Satoshi
  full_name: Masaki, Satoshi
  last_name: Masaki
- first_name: Jason
  full_name: Murphy, Jason
  last_name: Murphy
- first_name: Monica
  full_name: Visan, Monica
  id: 056daca0-b8d1-11f0-964f-f91054abf8ca
  last_name: Visan
citation:
  ama: Killip R, Masaki S, Murphy J, Vişan M. The radial mass-subcritical NLS in negative
    order Sobolev spaces. <i>Discrete and Continuous Dynamical Systems</i>. 2019;39(1):553-583.
    doi:<a href="https://doi.org/10.3934/dcds.2019023">10.3934/dcds.2019023</a>
  apa: Killip, R., Masaki, S., Murphy, J., &#38; Vişan, M. (2019). The radial mass-subcritical
    NLS in negative order Sobolev spaces. <i>Discrete and Continuous Dynamical Systems</i>.
    American Institute of Mathematical Sciences. <a href="https://doi.org/10.3934/dcds.2019023">https://doi.org/10.3934/dcds.2019023</a>
  chicago: Killip, Rowan, Satoshi Masaki, Jason Murphy, and Monica Vişan. “The Radial
    Mass-Subcritical NLS in Negative Order Sobolev Spaces.” <i>Discrete and Continuous
    Dynamical Systems</i>. American Institute of Mathematical Sciences, 2019. <a href="https://doi.org/10.3934/dcds.2019023">https://doi.org/10.3934/dcds.2019023</a>.
  ieee: R. Killip, S. Masaki, J. Murphy, and M. Vişan, “The radial mass-subcritical
    NLS in negative order Sobolev spaces,” <i>Discrete and Continuous Dynamical Systems</i>,
    vol. 39, no. 1. American Institute of Mathematical Sciences, pp. 553–583, 2019.
  ista: Killip R, Masaki S, Murphy J, Vişan M. 2019. The radial mass-subcritical NLS
    in negative order Sobolev spaces. Discrete and Continuous Dynamical Systems. 39(1),
    553–583.
  mla: Killip, Rowan, et al. “The Radial Mass-Subcritical NLS in Negative Order Sobolev
    Spaces.” <i>Discrete and Continuous Dynamical Systems</i>, vol. 39, no. 1, American
    Institute of Mathematical Sciences, 2019, pp. 553–83, doi:<a href="https://doi.org/10.3934/dcds.2019023">10.3934/dcds.2019023</a>.
  short: R. Killip, S. Masaki, J. Murphy, M. Vişan, Discrete and Continuous Dynamical
    Systems 39 (2019) 553–583.
das_tickbox: '1'
date_created: 2026-06-19T07:41:46Z
date_published: 2019-01-01T00:00:00Z
date_updated: 2026-06-22T11:06:54Z
day: '01'
doi: 10.3934/dcds.2019023
extern: '1'
external_id:
  arxiv:
  - '1804.06753'
intvolume: '        39'
issue: '1'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.1804.06753
month: '01'
oa: 1
oa_version: Preprint
page: 553-583
publication: Discrete and Continuous Dynamical Systems
publication_identifier:
  eissn:
  - 1553-5231
  issn:
  - 1078-0947
publication_status: published
publisher: American Institute of Mathematical Sciences
quality_controlled: '1'
scopus_import: '1'
status: public
title: The radial mass-subcritical NLS in negative order Sobolev spaces
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 39
year: '2019'
...
---
OA_place: repository
OA_type: green
_id: '22033'
abstract:
- lang: eng
  text: "We consider the defocusing energy-critical nonlinear Schrödinger\r\nequation
    with inverse-square potential iut = −∆u + a|x|^−2u + |u|^4u in three\r\nspace
    dimensions. We prove global well-posedness and scattering for a >− 1/4 + 1/25.
    We also carry out the variational analysis needed to treat the focusing case."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Rowan
  full_name: Killip, Rowan
  last_name: Killip
- first_name: Changxing
  full_name: Miao, Changxing
  last_name: Miao
- first_name: Monica
  full_name: Visan, Monica
  id: 056daca0-b8d1-11f0-964f-f91054abf8ca
  last_name: Visan
- first_name: Junyong
  full_name: Zhang, Junyong
  last_name: Zhang
- first_name: Jiqiang
  full_name: Zheng, Jiqiang
  last_name: Zheng
citation:
  ama: Killip R, Miao C, Vişan M, Zhang J, Zheng J. The energy-critical NLS with inverse-square
    potential. <i>Discrete and Continuous Dynamical Systems</i>. 2017;37(7):3831-3866.
    doi:<a href="https://doi.org/10.3934/dcds.2017162">10.3934/dcds.2017162</a>
  apa: Killip, R., Miao, C., Vişan, M., Zhang, J., &#38; Zheng, J. (2017). The energy-critical
    NLS with inverse-square potential. <i>Discrete and Continuous Dynamical Systems</i>.
    American Institute of Mathematical Sciences. <a href="https://doi.org/10.3934/dcds.2017162">https://doi.org/10.3934/dcds.2017162</a>
  chicago: Killip, Rowan, Changxing Miao, Monica Vişan, Junyong Zhang, and Jiqiang
    Zheng. “The Energy-Critical NLS with Inverse-Square Potential.” <i>Discrete and
    Continuous Dynamical Systems</i>. American Institute of Mathematical Sciences,
    2017. <a href="https://doi.org/10.3934/dcds.2017162">https://doi.org/10.3934/dcds.2017162</a>.
  ieee: R. Killip, C. Miao, M. Vişan, J. Zhang, and J. Zheng, “The energy-critical
    NLS with inverse-square potential,” <i>Discrete and Continuous Dynamical Systems</i>,
    vol. 37, no. 7. American Institute of Mathematical Sciences, pp. 3831–3866, 2017.
  ista: Killip R, Miao C, Vişan M, Zhang J, Zheng J. 2017. The energy-critical NLS
    with inverse-square potential. Discrete and Continuous Dynamical Systems. 37(7),
    3831–3866.
  mla: Killip, Rowan, et al. “The Energy-Critical NLS with Inverse-Square Potential.”
    <i>Discrete and Continuous Dynamical Systems</i>, vol. 37, no. 7, American Institute
    of Mathematical Sciences, 2017, pp. 3831–66, doi:<a href="https://doi.org/10.3934/dcds.2017162">10.3934/dcds.2017162</a>.
  short: R. Killip, C. Miao, M. Vişan, J. Zhang, J. Zheng, Discrete and Continuous
    Dynamical Systems 37 (2017) 3831–3866.
das_tickbox: '1'
date_created: 2026-06-19T07:42:59Z
date_published: 2017-07-01T00:00:00Z
date_updated: 2026-06-22T12:47:32Z
day: '01'
doi: 10.3934/dcds.2017162
extern: '1'
external_id:
  arxiv:
  - '1509.05822'
intvolume: '        37'
issue: '7'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.1509.05822
month: '07'
oa: 1
oa_version: Preprint
page: 3831-3866
publication: Discrete and Continuous Dynamical Systems
publication_identifier:
  eissn:
  - 1553-5231
  issn:
  - 1078-0947
publication_status: published
publisher: American Institute of Mathematical Sciences
quality_controlled: '1'
scopus_import: '1'
status: public
title: The energy-critical NLS with inverse-square potential
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 37
year: '2017'
...