[Inégalités de Keller–Lieb–Thirring pour des opérateurs de Schrödinger sur des cylindres]
Cette note est consacrée à des estimations spectrales de Keller–Lieb–Thirring pour des opérateurs de Schrödinger sur des cylindres infinis : la valeur absolue de l'état fondamental est bornée par une fonction d'une norme du potentiel. Il est montré que les potentiels optimaux de petite norme ne dépendent que d'une seule variable : il s'agit d'un résultat de symétrie. La preuve provient d'un argument de perturbation qui repose sur des résultats de rigidité récents pour des équations elliptiques non linéaires sur des cylindres. À l'inverse, les potentiels optimaux de grande norme qui ne dépendent que d'une seule variable sont instables : cela fournit un résultat de brisure de symétrie. La valeur optimale qui sépare les deux régimes est établie dans le cas du produit d'une sphère et d'une droite.
This note is devoted to Keller–Lieb–Thirring spectral estimates for Schrödinger operators on infinite cylinders: the absolute value of the ground state level is bounded by a function of a norm of the potential. Optimal potentials with small norms are shown to depend on a single variable: this is a symmetry result. The proof is a perturbation argument based on recent rigidity results for nonlinear elliptic equations on cylinders. Conversely, optimal single variable potentials with large norms must be unstable: this provides a symmetry breaking result. The optimal threshold between the two regimes is established in the case of the product of a sphere by a line.
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@article{CRMATH_2015__353_9_813_0,
author = {Dolbeault, Jean and Esteban, Maria J. and Loss, Michael},
title = {Keller{\textendash}Lieb{\textendash}Thirring inequalities for {Schr\"odinger} operators on cylinders},
journal = {Comptes Rendus. Math\'ematique},
pages = {813--818},
publisher = {Elsevier},
volume = {353},
number = {9},
year = {2015},
doi = {10.1016/j.crma.2015.06.018},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2015.06.018/}
}
TY - JOUR AU - Dolbeault, Jean AU - Esteban, Maria J. AU - Loss, Michael TI - Keller–Lieb–Thirring inequalities for Schrödinger operators on cylinders JO - Comptes Rendus. Mathématique PY - 2015 SP - 813 EP - 818 VL - 353 IS - 9 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2015.06.018/ DO - 10.1016/j.crma.2015.06.018 LA - en ID - CRMATH_2015__353_9_813_0 ER -
%0 Journal Article %A Dolbeault, Jean %A Esteban, Maria J. %A Loss, Michael %T Keller–Lieb–Thirring inequalities for Schrödinger operators on cylinders %J Comptes Rendus. Mathématique %D 2015 %P 813-818 %V 353 %N 9 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2015.06.018/ %R 10.1016/j.crma.2015.06.018 %G en %F CRMATH_2015__353_9_813_0
Dolbeault, Jean; Esteban, Maria J.; Loss, Michael. Keller–Lieb–Thirring inequalities for Schrödinger operators on cylinders. Comptes Rendus. Mathématique, Tome 353 (2015) no. 9, pp. 813-818. doi : 10.1016/j.crma.2015.06.018. http://www.numdam.org/articles/10.1016/j.crma.2015.06.018/
[1] On the Caffarelli–Kohn–Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions, Commun. Pure Appl. Math., Volume 54 (2001) no. 2, pp. 229-258
[2] Branches of non-symmetric critical points and symmetry breaking in nonlinear elliptic partial differential equations, Nonlinearity, Volume 27 (2014) no. 3, p. 435
[3] Spectral estimates on the sphere, Anal. PDE, Volume 7 (2014) no. 2, pp. 435-460
[4] Spectral properties of Schrödinger operators on compact manifolds: rigidity, flows, interpolation and spectral estimates, C. R. Acad. Sci. Paris, Ser. I, Volume 351 (2013) no. 11–12, pp. 437-440
[5] One-dimensional Gagliardo–Nirenberg–Sobolev inequalities: remarks on duality and flows, J. Lond. Math. Soc., Volume 90 (2014) no. 2, pp. 525-550
[6] Symmetry of extremals of functional inequalities via spectral estimates for linear operators, J. Math. Phys., Volume 53 (2012), p. 095204
[7] J. Dolbeault, M.J. Esteban, M. Loss, Rigidity versus symmetry breaking via nonlinear flows on cylinders and euclidean spaces, Preprint, 2015, . | HAL
[8] On the symmetry of extremals for the Caffarelli–Kohn–Nirenberg inequalities, Adv. Nonlinear Stud., Volume 9 (2009) no. 4, pp. 713-726
[9] J. Dolbeault, M. Kowalczyk, Uniqueness and rigidity in nonlinear elliptic equations, interpolation inequalities, and spectral estimates, Preprint, 2014, . | HAL
[10] Perturbation results of critical elliptic equations of Caffarelli–Kohn–Nirenberg type, J. Differ. Equ., Volume 191 (2003) no. 1, pp. 121-142
[11] Lower bounds and isoperimetric inequalities for eigenvalues of the Schrödinger equation, J. Math. Phys., Volume 2 (1961), pp. 262-266
[12] Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities (Lieb, E.; Simon, B.; Wightman, A., eds.), Essays in Honor of Valentine Bargmann, Princeton University Press, 1976, pp. 269-303
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