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Revista de Matemática Teoría y Aplicaciones

versión impresa ISSN 1409-2433

Rev. Mat vol.21 no.1 San José ene./jun. 2014

 

On the design of membranes with increasing fundamental frequency

Sobre el diseño de membranas con frecuencia fundamental creciente

Raúl B. González De Paz*+

*Dirección para correspondencia:

Abstract

By means of a relaxation approach, we study the shape design of a stiff inclusion with given area in a membrane in order to maximize its fundamental frequency. As an eigenvalue control problem, the fundamental frequency is a concave function of the control, which is not described by the membrane shape, but by an element in a function space. First order optimality conditions allow to describe the optimal shape by means of a free boundary value problem.

Keywords: variational methods for eigenvalues, shape optimization, free boundary value problems.

Resumen

Mediante un método de relajación, se estudia la forma de una inclusión rígida de área dada en una membrana de manera que se maximice su frecuencia fundamental. Analizado como un problema de control de valores propios, la frecuencia fundamental es una función cóncava del control, el cual no es descrito por la forma de la membrana, sino por un elemento de un espacio de funciones. Las condiciones de optimalidad de primer orden permiten describir la forma óptima mediante un problema de frontera libre.

Palabras clave: métodos variacionales para valores propios, optimización de forma, problemas de frontera libre.

Mathematics Subject Classification: 35J20, 35R35, 49R05, 49Q10.


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References

[1] Bourbaki, N. (1973) Espaces Vectoriels Topologiques. Hermann, Paris.         [ Links ]

[2] Buttazzo, G.; Dal Maso, G. (1993) “An Existence Result for a Class of Shape Optimization Problems”, Arch. Rat. Mech. Anal. 122: 183–195.         [ Links ]

[3] Castaing, Ch.; Valadier,M. (1977) Convex Analysis and Measurable Multifunctions. Lecture Notes in Mathematics 580, Springer-Verlag,Berlin-New York.         [ Links ]

[4] Céa, J.; Malanowski, K. (1970) “An example of a max-min problem in partial differential equations”, SIAM J. on Control and Optimization 8: 305–316.         [ Links ]

[5] Courant, R.; Hilbert, D. (1953) Methods of Mathematical Physics, vol I. Wiley-Interscience, New York,         [ Links ]

[6] Cox, S.; McLaughlin, J. (1990) “Extremal eigenvalue problems for composite membranes”, Appl. Math. and Optim. 22: 169–187.         [ Links ]

[7] Delfour, M. (1992) “Shape derivatives and differentiability of min-max”, in: M. Delfour & G. Sabidussi (Eds.) Proceedings NATO-Université de Montréal Seminar on Shape Optimization and Free Boundaries, Kluwer, Dordrecht.         [ Links ]

[8] Egnell, H. (1987) “Extremal properties of the first eigenvalue of a class of elliptic eigenvalue problems”, Annali Sc. Norm. Sup. di Pisa 14: 1–48.         [ Links ]

[9] Eppler, K. (2011) “On the shape gradient computation for elliptic eigenvalue problems”, Research Report (August 2011)DFG SPP1253-120, Universität Erlangen-Nürnberg.         [ Links ]

[10] Ekeland, I.; Temam, R. (1974) Analyse Convexe et Problemes Variationnelles. Dunod, Paris.         [ Links ]

[11] Gonzalez de Paz, R.B. (1982) “Sur un problème d’optimisation de domaine”, Numer. Funct. Anal. and Optimiz. 5: 173–197.         [ Links ]

[12] Gonzalez de Paz, R.B. (1994) “A relaxation approach applied to domain optimization”, SIAM J. on Control and Optimization 32: 154–169.         [ Links ]

[13] Gonzalez de Paz, R.B.; Tiihonen, T. (1994) “On a relaxation based numerical method por domain optimization”, in: M. Krizek, P. Neittaanmaki & R. Sternberg (Eds.) Finite Element Methods, Marcel Dekker, New York.         [ Links ]

[14] Harrel, E.M.; Kröger, P.; Kurata, K. 2001) “On the placement of an obstacle or a well so as to optimize the fundamental eigenvalue”, SIAM J. Math. Anal. 33(1): 240–259.         [ Links ]

[15] Henrot, A. (2006) Extremum Problems for Eigenvalues of Elliptic Operators. Birkhäuser Verlag, Basel-Boston-Berlin.         [ Links ]

[16] Jensen, R. (1980) “Boundary regularity for variational inequalities”, Indiana Univ. Math. Journal 29: 495–511.         [ Links ]

[17] Jouron, Cl. (1978) “Sur un problème d’optimisation ou la contrainte portée sur la fréquence fondamentale”, RAIR0-Analyse Numérique 12: 349–374.         [ Links ]

[18] Kawohl, B. (1986) “Geometrical properties of level sets of solutions to elliptic problems”, Proc. Symp. AMS in Pure Mathematics 45: 25–36.         [ Links ]

[19] Kinderlehrer, D.; Stampacchia, G. (1980) An Introduction to Variational Inequalities and their Applications. Academic Press, New York.         [ Links ]

[20] Miranda, C. (1970) Partial Differential Equations of Elliptic Type. Springer Verlag, Berlin-New York.         [ Links ]

[21] Necas, J. (1967) Les Méthodes Directes en Théorie des Equations Elliptiques. Masson, Paris.         [ Links ]

[22] Payne, L.; Weinberger, H. (1961) “Some isoperimetric inequalities for membrane frequencies and torsional rigidity”, J. on Math. Anal. and Appl. 2: 210–216.         [ Links ]

[23] Rousselet, B. (1979) “Optimal design and eigenvalue problems”, Proc. 8th, IFIP Conference on Optimization Techniques, Lect. Notes in Control and Informaction Sciences, Vol 6, Springer-Verlag, Berlin: 342–352.         [ Links ]

[24] Simon, J. (1980) “Differentiation with respect to the domain in boundary value problems”, Numer. Funct. Anal. and Optimiz. 2: 649–687.         [ Links ]

[25] Sokolowski, J.; Zolesio, J.P. (1992) Introduction to Shape Optimization, Shape Sensitivity Analysis. Springer Verlag, New York-Berlin.         [ Links ]

[26] Tahraoui, R. (1988) “Quelques remarques sur le contrôle des valeurs propres”, in: H. Brézis & J.L. Lions (Eds.) Nonlinear Partial Differential Equations and Their Applications, Collège de France Seminar, vol. VIII, Longman, Essex: 176–213.         [ Links ]

[27] Valadier, M. (1970) Quelques Contributions à l’Analyse Convexe. Thèse Doctorale, Université de Paris.         [ Links ]

[28] Valadier, M. (1963) Extension d’un algoritnme de Frank et Wolfe, Rev. Franc. de Rech. Operationelle 36: 251–253.         [ Links ]

[29] Zolesio, J.P. (1981) “Domain variational formulation for free boundary problems”, in: J. Céa & E. Haug (Eds.) Optimization of Distributed Parameter Systems, vol. II, Sijthoff and Noordhoff, Alphen aan den Rijn: 1152–1194.         [ Links ]

[30] Zolesio, J.P. (1981) “Semiderivatives of repeated eigenvalues”, in: J. Céa & E. Haug (Eds.) Optimization of Distributed Parameter Systems, vol. II, Sijthoff and Noordhoff, Alphen aan den Rijn: 1457–1473.         [ Links ]

*Correspondencia a:
Raúl B. González De Paz: Departamento de Matematicas, Universidad del Valle de Guatemala, Apdo. Postal 82, 01901 Guatemala, Guatemala. E-Mail: ragopa@ufm.edu
*Departamento de Matematicas, Universidad del Valle de Guatemala, Apdo. Postal 82, 01901 Guatemala, Guatemala. E-Mail: ragopa@ufm.edu

Received: 30/Jan/2013; Revised: 8/Nov/2013; Accepted: 15/Nov/2013

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