Asymptotic shape optimization for Riesz means of the Dirichlet Laplacian over convex domains

  • Simon Larson

    KTH Royal Institute of Technology, Stockholm, Sweden
Asymptotic shape optimization for Riesz means of the Dirichlet Laplacian over convex domains cover

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Abstract

For , a convex and bounded domain, we study the spectrum of the Dirichlet Laplacian on . For any and let denote any extremal set of the shape optimization problem

where is an admissible family of convex domains in . If and is a positive sequence tending to infinity we prove that is a bounded sequence, and hence contains a convergent subsequence. Under an additional assumption on we characterize the possible limits of such subsequences as minimizers of the perimeter among domains in of unit measure. For instance if is the set of all convex polygons with no more than faces, then converges, up to rotation and translation, to the regular -gon.

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Cite this article

Simon Larson, Asymptotic shape optimization for Riesz means of the Dirichlet Laplacian over convex domains. J. Spectr. Theory 9 (2019), no. 3, pp. 857–895

DOI 10.4171/JST/265