Shape analysis in membrane vibration

In order to characterize the domain Omega minimizing the normal stress on t he boundary of a membrane, we are concerned with the shape derivative of th e functional J(Omega) = integral(I)integral(partial derivative Omega) (part ial derivative y/partial derivative n)(2) g dx dt, where I is the time inte rval, 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 wit h respect to the domain. Eventually, we give a necessary condition of optim ality which relies heavily on the oriented distance function and its proper ties around the neighbourhood of the boundary. Copyright (C) 2000 John Wile y & Sons, Ltd.