Convex shape optimization for the least biharmonic Steklov eigenvalue

Simão Antunes, Pedro Ricardo; Gazzola, Filippo

ESAIM: Control, Optimisation and Calculus of Variations, Tome 19 (2013) no. 2, pp. 385-403.

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.

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