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Inequalities for the lowest magnetic Neumann eigenvalue

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Abstract

We study the ground-state energy of the Neumann magnetic Laplacian on planar domains. For a constant magnetic field, we consider the question whether the disc maximizes this eigenvalue for fixed area. More generally, we discuss old and new bounds obtained on this problem.

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Notes

  1. Note that in [13] the normal is directed inwards.

  2. In the case of the radial function, we use the same notation for the function and the corresponding 1D function.

  3. The authors use another lowest eigenvalue corresponding to a Laplacian on 2-forms satisfying specific boundary conditions but work in any dimension. Here we are in dimension 2 and identify 2-forms and functions.

  4. We thank B. Colbois for communicating to us these papers before publication.

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Acknowledgements

The discussion on this problem started at a nice meeting in Oberwolfach (December 2014) organized by Bonnaillie–Noël, Kovařík and Pankrashkin. We would like to thank M. van den Berg, D. Bucur, B. Colbois, M. Persson Sundqvist, K. Pankrashkin, N. Popoff and A. Savo for discussions around this problem. We also thank the referee for additional references.

Funding

Funding was provided by Det Frie Forskningsråd (Grant No. DFF-4181-00221).

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Authors and Affiliations

  1. Aarhus University, Ny Munkegade 118, 8000, Aarhus C, Denmark

    S. Fournais

  2. LMJL, Université de Nantes and CNRS, 2 rue de la Houssinière, 44322, Nantes Cedex, France

    B. Helffer

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  1. S. Fournais
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  2. B. Helffer
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Correspondence to S. Fournais.

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Fournais, S., Helffer, B. Inequalities for the lowest magnetic Neumann eigenvalue. Lett Math Phys 109, 1683–1700 (2019). https://doi.org/10.1007/s11005-018-01154-8

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  • DOI: https://doi.org/10.1007/s11005-018-01154-8

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