Volume 19, Number 2, April-June 2013
|Page(s)||385 - 403|
|Published online||10 January 2013|
Convex shape optimization for the least biharmonic Steklov eigenvalue
Departamento de Matemática, Universidade Lusófona de Humanidades e
Tecnologias, av. do Campo Grande
2 Grupo de Física Matemática da Universidade de Lisboa, Complexo Interdisciplinar, av. Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal
3 Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy
Revised: 13 February 2012
The least Steklov eigenvalue d1 for the biharmonic operator in bounded domains gives a bound for the positivity preserving property for the hinged plate problem, appears as a norm of a suitable trace operator, and gives the optimal constant to estimate the L2-norm of harmonic functions. These applications suggest to address the problem of minimizing d1 in suitable classes of domains. We survey the existing results and conjectures about this topic; in particular, the existence of a convex domain of fixed measure minimizing d1 is known, although the optimal shape is still unknown. We perform several numerical experiments which strongly suggest that the optimal planar shape is the regular pentagon. We prove the existence of a domain minimizing d1 also among convex domains having fixed perimeter and present some numerical results supporting the conjecture that, among planar domains, the disk is the minimizer.
Mathematics Subject Classification: 35P15 / 35J40 / 65N80
Key words: Biharmonic operator / least Steklov eigenvalue / shape optimization / numerical method of fundamental solutions
© EDP Sciences, SMAI, 2013
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