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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.
@article{COCV_2013__19_2_385_0,
author = {Sim\~ao Antunes, Pedro Ricardo and Gazzola, Filippo},
title = {Convex shape optimization for the least biharmonic {Steklov} eigenvalue},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {385--403},
publisher = {EDP-Sciences},
volume = {19},
number = {2},
year = {2013},
doi = {10.1051/cocv/2012014},
mrnumber = {3049716},
zbl = {1263.35171},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv/2012014/}
}
TY - JOUR AU - Simão Antunes, Pedro Ricardo AU - Gazzola, Filippo TI - Convex shape optimization for the least biharmonic Steklov eigenvalue JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2013 SP - 385 EP - 403 VL - 19 IS - 2 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/cocv/2012014/ DO - 10.1051/cocv/2012014 LA - en ID - COCV_2013__19_2_385_0 ER -
%0 Journal Article %A Simão Antunes, Pedro Ricardo %A Gazzola, Filippo %T Convex shape optimization for the least biharmonic Steklov eigenvalue %J ESAIM: Control, Optimisation and Calculus of Variations %D 2013 %P 385-403 %V 19 %N 2 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/cocv/2012014/ %R 10.1051/cocv/2012014 %G en %F COCV_2013__19_2_385_0
Simão Antunes, Pedro Ricardo; Gazzola, Filippo. Convex shape optimization for the least biharmonic Steklov eigenvalue. ESAIM: Control, Optimisation and Calculus of Variations, Tome 19 (2013) no. 2, pp. 385-403. doi: 10.1051/cocv/2012014
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