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Spectral Gap for Stable Process on Convex Planar Double Symmetric Domains

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Abstract

We study the semigroup of the symmetric α-stable process in bounded domains in R 2. We obtain a variational formula for the spectral gap, i.e. the difference between two first eigenvalues of the generator of this semigroup. This variational formula allows us to obtain lower bound estimates of the spectral gap for convex planar domains which are symmetric with respect to both coordinate axes. For rectangles, using “midconcavity” of the first eigenfunction (Bañuelos et al., Potential Anal. 24(3): 205–221, 2006), we obtain sharp upper and lower bound estimates of the spectral gap.

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Authors and Affiliations

  1. Institute of Mathematics and Computer Science, Wrocław University of Technology, Wybrzeże Wyspiańskiego 27, 50-370, Wrocław, Poland

    Bartłomiej Dyda & Tadeusz Kulczycki

Authors
  1. Bartłomiej Dyda
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  2. Tadeusz Kulczycki
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Correspondence to Tadeusz Kulczycki.

Additional information

The first named author was supported by KBN grant 1 P03A 026 29 and RTN Harmonic Analysis and Related Problems, contract HPRN-CT-2001-00273-HARP

The second named author was supported by KBN grant 1 P03A 020 28 and RTN Harmonic Analysis and Related Problems, contract HPRN-CT-2001-00273-HARP.

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Dyda, B., Kulczycki, T. Spectral Gap for Stable Process on Convex Planar Double Symmetric Domains. Potential Anal 27, 101–132 (2007). https://doi.org/10.1007/s11118-007-9046-4

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  • DOI: https://doi.org/10.1007/s11118-007-9046-4

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