Inequalities involving Aharonov-Bohm magnetic potentials in dimensions 2 and 3

Denis Bonheure (1) , Jean Dolbeault (2) , Maria J. Esteban (2) , Ari Laptev (3) , Michael Loss (4)

1 Département de mathématiques Université Libre de Bruxelles
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
3 Department of Mathematics [Imperial College London]
4 School of Mathematics - Georgia Institute of Technology

This paper is devoted to a collection of results on nonlinear interpolation inequalities associated with Schrödinger operators involving Aharonov-Bohm magnetic potentials, and to some consequences. As symmetry plays an important role for establishing optimality results, we shall consider various cases corresponding to a circle, a two-dimensional sphere or a two-dimensional torus, and also the Euclidean spaces of dimensions two and three. Most of the results are new and we put the emphasis on the methods, as very little is known on symmetry, rigidity and optimality in presence of a magnetic field. The most spectacular applications are new magnetic Hardy inequalities in dimensions 2 and 3.

Mots clés: Aharonov-Bohm magnetic potential; radial symmetry; cylindrical symmetry; symmetry breaking; magnetic Hardy inequality; magnetic interpolation inequality; optimal constants; magnetic Schrödinger operator; magnetic Keller-Lieb-Thirring inequality; magnetic rings

Domaines: Equations aux dérivées partielles [math.AP]; Physique mathématique [math-ph]

Denis Bonheure, Jean Dolbeault, Maria J. Esteban, Ari Laptev, Michael Loss. Inequalities involving Aharonov-Bohm magnetic potentials in dimensions 2 and 3. 2019. ⟨hal-02021174v1⟩