Branches of non-symmetric critical points and symmetry breaking in nonlinear elliptic partial differential equations

Dolbeault, Jean; Esteban, Maria J.

doi:10.1088/0951-7715/27/3/435

Cited by 11 publications

(18 citation statements)

References 16 publications

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“…See [29] for the existence of the curve p → α(p), [23,24] for various results on symmetry in a larger class of inequalities, and [28] for Property (ii) in Proposition 19. Numerical computations of the branches of non-radial optimal functions and formal asymptotic expansions at the bifurcation point have been collected in [26,36]. The paper [30] deals with the special case of dimension d = 2 and contains Property (iii) in Proposition 19, which can be rephrased as follows: the region of radial symmetry contains the region corresponding to a ≥ α(p) and b ≥ β(p), and the parametric curve p → (α(p), β(p)) converges to 0 as p → 2 * = ∞ tangentially to the axis b = 0.…”

Section: Appendix B Poincaré Inequality and Stereographic Projectionmentioning

confidence: 99%

Sobolev and Hardy–Littlewood–Sobolev inequalities

Dolbeault

Jankowiak

2014

Journal of Differential Equations

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This paper is devoted to improvements of Sobolev and Onofri inequalities. The additional terms involve the dual counterparts, i.e. Hardy-Littlewood-Sobolev type inequalities. The Onofri inequality is achieved as a limit case of Sobolev type inequalities. Then we focus our attention on the constants in our improved Sobolev inequalities, that can be estimated by completion of the square methods. Our estimates rely on nonlinear flows and spectral problems based on a linearization around optimal Aubin-Talenti functions.

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Section: Appendix B Poincaré Inequality and Stereographic Projectionmentioning

confidence: 99%

Sobolev and Hardy–Littlewood–Sobolev inequalities

Dolbeault

Jankowiak

2014

Journal of Differential Equations

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“…For any w ∈ H 1 (R × S 1 ), let us define Proof. The existence of a minimizer is obtained as in the standard case corresponding to ν = 1 and we refer to [8]…”

Section: Magnetic Hardy and Non-magnetic Hardy-sobolev Inequalitiesmentioning

confidence: 99%

Symmetry Results in Two-Dimensional Inequalities for Aharonov–Bohm Magnetic Fields

Bonheure

Dolbeault

Esteban

et al. 2019

Commun. Math. Phys.

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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-Thirring 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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“…Such symmetry breaking issues are however known to be difficult: see for instance [23] for a discussion of a related problem.…”

Section: Weighted Moser-trudinger-onofri Inequalities On the Two-dimementioning

confidence: 99%

Onofri inequalities and rigidity results

Dolbeault¹,

Esteban²,

Jankowiak³

2017

Discrete &Amp; Continuous Dynamical Systems - A

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This paper is devoted to the Moser-Trudinger-Onofri inequality on smooth compact connected Riemannian manifolds. We establish a rigidity result for the Euler-Lagrange equation and deduce an estimate of the optimal constant in the inequality on two-dimensional closed Riemannian manifolds. Compared to existing results, we provide a non-local criterion which is well adapted to variational methods, introduce a nonlinear flow along which the evolution of a functional related with the inequality is monotone and get an integral remainder term which allows us to discuss optimality issues. As an important application of our method, we also consider the non-compact case of the Moser-Trudinger-Onofri inequality on the two-dimensional Euclidean space, with weights. The standard weight is the one that is computed when projecting the two-dimensional sphere using the stereographic projection, but we also give more general results which are of interest, for instance, for the Keller-Segel model in chemotaxis.2010 Mathematics Subject Classification. Primary: 58J35, 58J05, 53C21; Secondary: 35J60, 35K55.

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Branches of non-symmetric critical points and symmetry breaking in nonlinear elliptic partial differential equations

Cited by 11 publications

References 16 publications

Sobolev and Hardy–Littlewood–Sobolev inequalities

Sobolev and Hardy–Littlewood–Sobolev inequalities

Symmetry Results in Two-Dimensional Inequalities for Aharonov–Bohm Magnetic Fields

Onofri inequalities and rigidity results

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