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.
Mots clés : biharmonic operator, least Steklov eigenvalue, shape optimization, numerical method of fundamental solutions
@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://www.numdam.org/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://www.numdam.org/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://www.numdam.org/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. http://www.numdam.org/articles/10.1051/cocv/2012014/
[1] On the choice of source points in the method of fundamental solutions. Eng. Anal. Bound. Elem. 33 (2009) 1348-1361. | MR | Zbl
,[2] The method of fundamental solutions applied to the calculation of eigenfrequencies and eigenmodes of 2D simply connected shapes. Comput. Mater. Cont. 2 (2005) 251-266.
and ,[3] The method of fundamental solutions applied to the calculation of eigensolutions for 2D plates. Int. J. Numer. Methods Eng. 77 (2008) 177-194. | MR | Zbl
and ,[4] A numerical study of the spectral gap. J. Phys. A Math. Theor. 5 (2008) 055201. | MR | Zbl
and ,[5] On the range of the first two Dirichlet and Neumann eigenvalues of the Laplacian. Proc. R. Soc. London Ser. A Math. Phys. Eng. Sci. 467 (2011) 1577-1603. | MR | Zbl
and ,[6] On the lift-off constant for elastically supported plates. Proc. Amer. Math. Soc. 132 (2004) 2951-2958. | MR | Zbl
, and ,[7] Positivity preserving property for a class of biharmonic elliptic problems. J. Differ. Equ. 320 (2006) 1-23. | MR | Zbl
, and ,[8] The first biharmonic Steklov eigenvalue : positivity preserving and shape optimization. Milan J. Math. 79 (2011) 247-258. | MR | Zbl
and ,[9] On the first eigenvalue of a fourth order Steklov problem. Calc. Var. 35 (2009) 103-131. | MR | Zbl
, and ,[10] Minimum problems over sets of concave functions and related questions. Math. Nachr. 173 (1995) 71-89. | MR | Zbl
, and ,[11] On a fourth order Steklov eigenvalue problem. Analysis 25 (2005) 315-332. | MR | Zbl
, and ,[12] Su un principio di dualità per talune formole di maggiorazione relative alle equazioni differenziali. Atti. Accut. Naz. Lincei 19 (1955) 411-418. | MR | Zbl
,[13] Die randwert und eigenwertprobleme aus der theorie der elastischen platten. Math. Ann. 98 (1927) 205-247. | JFM
,[14] On positivity for the biharmonic operator under Steklov boundary conditions. Arch. Ration. Mech. Anal. 188 (2008) 399-427. | MR | Zbl
and ,[15] Polyharmonic boundary value problems. Lect. Notes Math. 1991 (2010). | MR | Zbl
, and ,[16] On “anti”-eigenvalues for elliptic systems and a question of McKenna and Walter. Indiana Univ. Math. J. 51 (2002) 1023-1040. | MR | Zbl
and ,[17] Über das gleichgewicht und die bewegung einer elastischen scheibe. J. Reine Angew. Math. 40 (1850) 51-88. | Zbl
,[18] Remarks on a Stekloff eigenvalue problem. SIAM J. Numer. Anal. 9 (1972) 1-5. | MR | Zbl
,[19] Foam structures with a negative Poisson's ratio. Science 235 (1987) 1038-1040.
,[20] Problèmes aux limites non homogènes et applications. Travaux et Recherches Mathématiques 3 (1970). | Zbl
and ,[21] The Weyl-type asymptotic formula for biharmonic Steklov eigenvalues on Riemannian manifolds. Adv. Math. 228 (2011) 2162-2217. | MR | Zbl
,[22] A treatise on the mathematical theory of elasticity, 4th edition. Cambridge Univ. Press (1927). | JFM | Zbl
,[23] Nonlinear oscillations in a suspension bridge. Arch. Ration. Mech. Anal. 98 (1987) 167-177. | MR | Zbl
and ,[24] On the positivity preserving property of hinged plates. SIAM J. Math. Anal. 41 (2009) 2031-2037. | MR | Zbl
and ,[25] Bounds for the maximum stress in the Saint Venant torsion problem. Special issue presented to Professor Bibhutibhusan Sen on the occasion of his seventieth birthday, Part I. Indian J. Mech. Math. (1968/1969) 51-59. | MR
,[26] Some isoperimetric inequalities for harmonic functions. SIAM J. Math. Anal. 1 (1970) 354-359. | MR | Zbl
,[27] Convex bodies : the Brunn-Minkowski theory. Cambridge Univ. Press (1993). | MR | Zbl
,[28] The coupled equation approach to the numerical solution of the biharmonic equation by finite differences I. SIAM J. Numer. Anal. 5 (1968) 323-339. | MR | Zbl
,[29] The coupled equation approach to the numerical solution of the biharmonic equation by finite differences II. SIAM J. Numer. Anal. 7 (1970) 104-111. | MR | Zbl
,[30] Sur les problèmes fondamentaux de la physique mathématique. Ann. Sci. Éc. Norm. Sup. 19 (1902) 191-259; 455-490. | JFM | Numdam
,[31] Wikipedia, the Free Encyclopedia, available on http://en.wikipedia.org/wiki/Reuleaux♯triangle♯Reuleaux−polygons
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