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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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