Convex shape optimization for the least biharmonic Steklov eigenvalue

¹ Departamento de Matemática, Universidade Lusófona de Humanidades e Tecnologias, av. do Campo Grande 376, 1749-024 Lisboa, portugal
pant@cii.fc.ul.pt
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
filippo.gazzola@polimi.it

Received: 10 November 2011
Revised: 13 February 2012

Abstract

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.