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An indefinite Laplacian on a rectangle

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Abstract

In this note, we investigate the nonelliptic differential expression

$$A = - div\operatorname{sgn} \nabla $$

on a rectangular domain Ω in the plane. The seemingly simple problem of associating a self-adjoint operator with the differential expression A in L2(Ω) is solved here. Such indefinite Laplacians arise in mathematical models of metamaterials characterized by negative electric permittivity and/or negative magnetic permeability.

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Authors and Affiliations

  1. Institut für Numerische Mathematik, Technische Universität Graz, Steyrergasse 30, 8010, Graz, Austria

    Jussi Behrndt

  2. Department of Mathematics Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Trojanova 13, 12000, Prague 2, Czech Republic

    David Krejčiřík

Authors
  1. Jussi Behrndt
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  2. David Krejčiřík
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Corresponding author

Correspondence to Jussi Behrndt.

Additional information

This work is supported by the Austrian Science Fund (FWF), project P 25162-N26, Czech project RVO61389005 and the GACR grant No. 14-06818S.

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Behrndt, J., Krejčiřík, D. An indefinite Laplacian on a rectangle. JAMA 134, 501–522 (2018). https://doi.org/10.1007/s11854-018-0015-1

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  • DOI: https://doi.org/10.1007/s11854-018-0015-1

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