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. Ver contenido disponible en pdf
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*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