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Steklov Eigenvalues and Quasiconformal Maps of Simply Connected Planar Domains

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Abstract

We investigate isoperimetric upper bounds for sums of consecutive Steklov eigenvalues of planar domains. The normalization involves the perimeter and scale-invariant geometric factors which measure deviation of the domain from roundness. We prove sharp upper bounds for both starlike and simply connected domains for a large collection of spectral functionals including partial sums of the zeta function and heat trace. The proofs rely on a special class of quasiconformal mappings.

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Authors and Affiliations

  1. Department de Mathematiques et Statistique, Université Laval, Quebec, QC, Canada

    A. Girouard

  2. Department of Mathematics, University of Illinois, Urbana, IL, 61801, USA

    R. S. Laugesen

  3. Department of Mathematics, University of Oregon, Eugene, OR, 97403, USA

    B. A. Siudeja

Authors
  1. A. Girouard
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  2. R. S. Laugesen
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  3. B. A. Siudeja
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Correspondence to B. A. Siudeja.

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Communicated by G. Dal Maso

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Girouard, A., Laugesen, R.S. & Siudeja, B.A. Steklov Eigenvalues and Quasiconformal Maps of Simply Connected Planar Domains. Arch Rational Mech Anal 219, 903–936 (2016). https://doi.org/10.1007/s00205-015-0912-8

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  • DOI: https://doi.org/10.1007/s00205-015-0912-8

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