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Stability of spectral partitions and the Dirichlet-to-Neumann map

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Abstract

The oscillation of a Laplacian eigenfunction gives a great deal of information about the manifold on which it is defined. This oscillation can be encoded in the nodal deficiency, an important geometric quantity that is notoriously hard to compute, or even estimate. Here we compare two recently obtained formulas for the nodal deficiency, one in terms of an energy functional on the space of equipartitions of the manifold, and the other in terms of a two-sided Dirichlet-to-Neumann map defined on the nodal set. We relate these two approaches by giving an explicit formula for the Hessian of the equipartition energy in terms of the Dirichlet-to-Neumann map. This allows us to compute Hessian eigenfunctions, and hence directions of steepest descent, for the equipartition energy in terms of the corresponding Dirichlet-to-Neumann eigenfunctions. Our results do not assume bipartiteness, and hence are relevant to the study of spectral minimal partitions.

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Notes

  1. It is an immediate consequence of our main result that (3) also holds for non-simple eigenvalues. In this case the Hessian is degenerate, with nullity determined by the multiplicity of the eigenvalue, as in (8).

  2. The non-smooth case is more involved and will be treated elsewhere; see the discussion at the end of Sect. 2.

  3. Throughout the paper, all integrals are with respect to the Riemannian volume measure on M, or the induced surface measure on \(\Sigma \); we do not indicate the measure explicitly since it will always be clear from the context.

  4. The smooth structure does not depend on the choice of unit normal, so we may assume that this is the same \(\nu \) that appears in the statement of Theorem 3.

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Acknowledgements

The authors thank Ram Band, Sebastian Egger, Bernard Helffer, Peter Kuchment, and Mikael Persson Sundqvist for helpful comments and discussions. G.B. acknowledges the support of NSF Grant DMS-1815075. Y.C. was supported by the Alfred P. Sloan Foundation and NSF CAREER Grant DMS-2045494 and DMS-1900519. G.C. acknowledges the support of NSERC grant RGPIN-2017-04259. J.L.M. acknowledges support from the NSF through NSF CAREER Grant DMS-1352353 and NSF grant DMS-1909035. The authors are grateful to the AIM SQuaRE program for hosting them and supporting the initiation of this project.

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Authors and Affiliations

  1. Department of Mathematics, Texas A &M University, College Station, TX, 77843-3368, USA

    G. Berkolaiko

  2. Department of Mathematics, University of North Carolina at Chapel Hill, Phillips Hall, Chapel Hill, NC, 27599, USA

    Y. Canzani & J. L. Marzuola

  3. Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL, A1C 5S7, Canada

    G. Cox

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  1. G. Berkolaiko
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  2. Y. Canzani
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  3. G. Cox
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  4. J. L. Marzuola
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Correspondence to G. Cox.

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Communicated by S. Terracini.

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Berkolaiko, G., Canzani, Y., Cox, G. et al. Stability of spectral partitions and the Dirichlet-to-Neumann map. Calc. Var. 61, 203 (2022). https://doi.org/10.1007/s00526-022-02311-7

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