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Symmetry results in two-dimensional inequalities for Aharonov-Bohm magnetic fields

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HS-magnetic-sharp.pdf (394.5Kb)
Date
2019
Publisher city
Paris
Link to item file
https://hal.archives-ouvertes.fr/hal-02003872
Dewey
Analyse
Sujet
Hardy-Sobolev inequalities; Caffarelli-Kohn-Nirenberg inequalities; magnetic rings; magnetic Schrödinger operator; Aharonov-Bohm magnetic potential; radial symmetry; symmetry breaking; magnetic Hardy-Sobolev inequality; magnetic interpolation inequality; optimal constants
Journal issue
Communications in Mathematical Physics
Publication date
2019
Article pages
17
Publisher
Springer
DOI
http://dx.doi.org/10.1007/s00220-019-03560-y
URI
https://basepub.dauphine.fr/handle/123456789/18555
Collections
  • CEREMADE : Publications
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Author
Bonheure, Denis
Dolbeault, Jean
Esteban, Maria J.
Laptev, Ari
Loss, Michael
Type
Article accepté pour publication ou publié
Abstract (EN)
This paper is devoted to the symmetry and symmetry breaking properties of a two-dimensional magnetic Schrödinger operator involving an Aharonov-Bohm magnetic vector potential. We investigate the symmetry properties of the optimal potential for the corresponding magnetic Keller-Lieb-Thir-ring inequality. We prove that this potential is radially symmetric if the intensity of the magnetic field is below an explicit threshold, while symmetry is broken above a second threshold corresponding to a higher magnetic field. The method relies on the study of the magnetic kinetic energy of the wave function and amounts to study the symmetry properties of the optimal functions in a magnetic Hardy-Sobolev interpolation inequality. We give a quantified range of symmetry by a non-perturbative method. To establish the symmetry breaking range, we exploit the coupling of the phase and of the modulus and also obtain a quantitative result.

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