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Convex shape optimization for the least biharmonic Stekloveigenvalue

Published online by Cambridge University Press:  10 January 2013

Pedro Ricardo Simão Antunes and
Filippo Gazzola
Pedro Ricardo Simão Antunes
Affiliation:
Departamento de Matemática, Universidade Lusófona de Humanidades e Tecnologias, av. do Campo Grande 376, 1749-024 Lisboa, portugal. pant@cii.fc.ul.ptGrupo de Física Matemática da Universidade de Lisboa, Complexo Interdisciplinar, av. Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal
Filippo Gazzola
Affiliation:
Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy; filippo.gazzola@polimi.it

Abstract

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The least Steklov eigenvalue d1 for the biharmonic operatorin bounded domains gives a bound for the positivity preserving property for the hingedplate problem, appears as a norm of a suitable trace operator, and gives the optimalconstant to estimate the L2-norm of harmonic functions. Theseapplications suggest to address the problem of minimizing d1in suitable classes of domains. We survey the existing results and conjectures about thistopic; in particular, the existence of a convex domain of fixed measure minimizingd1 is known, although the optimal shape is still unknown. Weperform several numerical experiments which strongly suggest that the optimal planar shapeis the regular pentagon. We prove the existence of a domain minimizingd1 also among convex domains having fixed perimeter andpresent some numerical results supporting the conjecture that, among planar domains, thedisk is the minimizer.

Keywords

Biharmonic operatorleast Steklov eigenvalueshape optimizationnumerical method of fundamental solutions

Information

Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 19 , Issue 2 , April 2013, pp. 385 - 403
Copyright
© EDP Sciences, SMAI, 2013

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