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Asymptotic shape optimization for Riesz means of the Dirichlet Laplacian over convex domains
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).ORCID iD: 0000-0002-0057-8211
2019 (English)In: Journal of Spectral Theory, ISSN 1664-039X, E-ISSN 1664-0403, Vol. 9, no 3, p. 857-895Article in journal (Refereed) Published
Abstract [en]

For Omega subset of R-n, a convex and bounded domain, we study the spectrum of -Delta(Omega) the Dirichlet Laplacian on Omega. For Lambda >= 0 and gamma >= 0 let Omega(Lambda,gamma)(A) denote any extremal set of the shape optimization problem sup {Tr(-Delta(Omega) - Lambda)(gamma) : Omega is an element of .A, vertical bar Omega vertical bar = 1}, where A is an admissible family of convex domains in R-n. If gamma >= 1 and (1) and {Lambda(j)}(j >= 1) is a positive sequence tending to infinity we prove that {Omega Lambda j , gamma(A)}(j >= 1) is a bounded sequence, and hence contains a convergent subsequence. Under an additional assumption on A we characterize the possible limits of such subsequences asminimizers of the perimeter among domains in A of unit measure. For instance if A is the set of all convex polygons with no more than m faces, then Omega(Lambda,gamma) converges, up to rotation and translation, to the regular m-gon.

Place, publisher, year, edition, pages
EUROPEAN MATHEMATICAL SOC , 2019. Vol. 9, no 3, p. 857-895
Keywords [en]
Shape optimization, Riesz eigenvalue means, eigenvalue sums, Dirichlet-Laplace operator, Weyl asymptotics, convexity
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-261050DOI: 10.4171/JST/265ISI: 000484709400004Scopus ID: 2-s2.0-85071167293OAI: oai:DiVA.org:kth-261050DiVA, id: diva2:1356368
Note

QC 20191001

Available from: 2019-10-01 Created: 2019-10-01 Last updated: 2019-10-01Bibliographically approved

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Larson, Simon

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