Hostname: page-component-76fb5796d-vvkck Total loading time: 0 Render date: 2024-04-25T16:10:27.756Z Has data issue: false hasContentIssue false
  • English
  • Français

Convex shape optimization for the least biharmonic Steklov eigenvalue

Published online by Cambridge University Press:  10 January 2013

Pedro Ricardo Simão Antunes  and
Filippo Gazzola
Pedro Ricardo Simão Antunes
Affiliation:
Departamento de Matemática, Universidade Lusófona de Humanidades e Tecnologias, av. do Campo Grande 376, 1749-024 Lisboa, portugal. pant@cii.fc.ul.pt Grupo de Física Matemática da Universidade de Lisboa, Complexo Interdisciplinar, av. Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal
Filippo Gazzola
Affiliation:
Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy; filippo.gazzola@polimi.it
Get access

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.

Keywords

Biharmonic operatorleast Steklov eigenvalueshape optimizationnumerical method of fundamental solutions
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 19 , Issue 2 , April 2013 , pp. 385 - 403
Copyright
© EDP Sciences, SMAI, 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alves, C.J.S., On the choice of source points in the method of fundamental solutions. Eng. Anal. Bound. Elem. 33 (2009) 1348–1361. Google Scholar
Alves, C.J.S. and Antunes, P.R.S., 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. Google Scholar
Alves, C.J.S. and Antunes, P.R.S., The method of fundamental solutions applied to the calculation of eigensolutions for 2D plates. Int. J. Numer. Methods Eng. 77 (2008) 177–194. Google Scholar
Antunes, P. and Freitas, P., A numerical study of the spectral gap. J. Phys. A Math. Theor. 5 (2008) 055201. Google Scholar
Antunes, P.R.S. and Henrot, A., 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. Google Scholar
Bass, R.F., Horák, J. and McKenna, P.J., On the lift-off constant for elastically supported plates. Proc. Amer. Math. Soc. 132 (2004) 2951–2958. Google Scholar
Berchio, E., Gazzola, F. and Mitidieri, E., Positivity preserving property for a class of biharmonic elliptic problems. J. Differ. Equ. 320 (2006) 1–23. Google Scholar
Bucur, D. and Gazzola, F., The first biharmonic Steklov eigenvalue : positivity preserving and shape optimization. Milan J. Math. 79 (2011) 247–258. Google Scholar
Bucur, D., Ferrero, A. and Gazzola, F., On the first eigenvalue of a fourth order Steklov problem. Calc. Var. 35 (2009) 103–131. Google Scholar
Buttazzo, G., Ferone, V. and Kawohl, B., Minimum problems over sets of concave functions and related questions. Math. Nachr. 173 (1995) 71–89. Google Scholar
Ferrero, A., Gazzola, F. and Weth, T., On a fourth order Steklov eigenvalue problem. Analysis 25 (2005) 315–332. Google Scholar
Fichera, G., Su un principio di dualità per talune formole di maggiorazione relative alle equazioni differenziali. Atti. Accut. Naz. Lincei 19 (1955) 411–418. Google Scholar
Friedrichs, K., Die randwert und eigenwertprobleme aus der theorie der elastischen platten. Math. Ann. 98 (1927) 205–247. Google Scholar
Gazzola, F. and Sweers, G., On positivity for the biharmonic operator under Steklov boundary conditions. Arch. Ration. Mech. Anal. 188 (2008) 399–427. Google Scholar
Gazzola, F., Grunau, H.C. and Sweers, G., Polyharmonic boundary value problems. Lect. Notes Math. 1991 (2010). Google Scholar
Kawohl, B. and Sweers, G., On “anti”-eigenvalues for elliptic systems and a question of McKenna and Walter. Indiana Univ. Math. J. 51 (2002) 1023–1040. Google Scholar
Kirchhoff, G.R., Über das gleichgewicht und die bewegung einer elastischen scheibe. J. Reine Angew. Math. 40 (1850) 51–88. Google Scholar
Kuttler, J.R., Remarks on a Stekloff eigenvalue problem. SIAM J. Numer. Anal. 9 (1972) 1–5. Google Scholar
Lakes, R.S., Foam structures with a negative Poisson’s ratio. Science 235 (1987) 1038–1040. Google ScholarPubMed
Lions, J.L. and Magenes, E., Problèmes aux limites non homogènes et applications. Travaux et Recherches Mathématiques 3 (1970). Google Scholar
Liu, G., The Weyl-type asymptotic formula for biharmonic Steklov eigenvalues on Riemannian manifolds. Adv. Math. 228 (2011) 2162–2217. Google Scholar
A.E.H. Love, A treatise on the mathematical theory of elasticity, 4th edition. Cambridge Univ. Press (1927).
McKenna, P.J. and Walter, W., Nonlinear oscillations in a suspension bridge. Arch. Ration. Mech. Anal. 98 (1987) 167–177. Google Scholar
Parini, E. and Stylianou, A., On the positivity preserving property of hinged plates. SIAM J. Math. Anal. 41 (2009) 2031–2037. Google Scholar
L.E. Payne, 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.
Payne, L.E., Some isoperimetric inequalities for harmonic functions. SIAM J. Math. Anal. 1 (1970) 354–359. Google Scholar
R. Schneider, Convex bodies : the Brunn-Minkowski theory. Cambridge Univ. Press (1993).
Smith, J., The coupled equation approach to the numerical solution of the biharmonic equation by finite differences I. SIAM J. Numer. Anal. 5 (1968) 323–339. Google Scholar
Smith, J., The coupled equation approach to the numerical solution of the biharmonic equation by finite differences II. SIAM J. Numer. Anal. 7 (1970) 104–111. Google Scholar
Stekloff, W., Sur les problèmes fondamentaux de la physique mathématique. Ann. Sci. Éc. Norm. Sup. 19 (1902) 191–259; 455–490. Google Scholar
Wikipedia, the Free Encyclopedia, available on http://en.wikipedia.org/wiki/Reuleaux♯triangle♯Reuleaux−polygons