Shape analysis in membrane vibration

In order to characterize the domain Ω minimizing the normal stress on the boundary of a membrane, we are concerned with the shape derivative of the functional \def\d{\,{\rm d}}$J(\Omega)=\int_I\int_{\partial\Omega}(\partial y/\partial n)^2\,g\d x\d t$ , where I is the time interval, y is the solution to the wave equation and g a weight coefficient. We first recall some results on the transformation of domains and investigate the shape derivative of the state. Then we compute the derivative of J with respect to the domain. Eventually, we give a necessary condition of optimality which relies heavily on the oriented distance function and its properties around the neighbourhood of the boundary. Copyright © 2000 John Wiley & Sons, Ltd.