Convex shape optimization for the least biharmonic Steklov eigenvalue
ESAIM: Control, Optimisation and Calculus of Variations, Volume 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.

DOI: 10.1051/cocv/2012014
Classification: 35P15, 35J40, 65N80
Keywords: biharmonic operator, least Steklov eigenvalue, shape optimization, numerical method of fundamental solutions
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     title = {Convex shape optimization for the least biharmonic {Steklov} eigenvalue},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {385--403},
     publisher = {EDP-Sciences},
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     mrnumber = {3049716},
     zbl = {1263.35171},
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     url = {http://www.numdam.org/articles/10.1051/cocv/2012014/}
}
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Simão Antunes, Pedro Ricardo; Gazzola, Filippo. Convex shape optimization for the least biharmonic Steklov eigenvalue. ESAIM: Control, Optimisation and Calculus of Variations, Volume 19 (2013) no. 2, pp. 385-403. doi : 10.1051/cocv/2012014. http://www.numdam.org/articles/10.1051/cocv/2012014/

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