Convex shape optimization for the least biharmonic Steklov eigenvalue

Pedro Ricardo Simão Antunes; Filippo Gazzola

ESAIM: Control, Optimisation and Calculus of Variations (2013)

  • Volume: 19, Issue: 2, page 385-403
  • ISSN: 1292-8119

Abstract

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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.

How to cite

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Simão Antunes, Pedro Ricardo, and Gazzola, Filippo. "Convex shape optimization for the least biharmonic Steklov eigenvalue." ESAIM: Control, Optimisation and Calculus of Variations 19.2 (2013): 385-403. <http://eudml.org/doc/272917>.

@article{SimãoAntunes2013,
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.},
author = {Simão Antunes, Pedro Ricardo, Gazzola, Filippo},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {biharmonic operator; least Steklov eigenvalue; shape optimization; numerical method of fundamental solutions},
language = {eng},
number = {2},
pages = {385-403},
publisher = {EDP-Sciences},
title = {Convex shape optimization for the least biharmonic Steklov eigenvalue},
url = {http://eudml.org/doc/272917},
volume = {19},
year = {2013},
}

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
PB - EDP-Sciences
VL - 19
IS - 2
SP - 385
EP - 403
AB - 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.
LA - eng
KW - biharmonic operator; least Steklov eigenvalue; shape optimization; numerical method of fundamental solutions
UR - http://eudml.org/doc/272917
ER -

References

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