Shape Sensitivity Analysis via Min Max Differentiability

Abstract

The object of this paper is twofold. We introduce a new theorem on the differentiability of a Min Max with respect to a parameter and we show how such a theorem can be applied to compute the material derivative in shape sensitivity analysis problems. We consider the Min Max of a functional which is parametrized by t. We show that, under appropriate conditions, the derivative of the Min Max with respect to t is the Min Max with respect to the points solution of the Min Max problem of the derivative of the original functional with respect to t. To illustrate the use of this theorem, we apply it to the control of an elliptic equation with a nondifferentiable observation and to shape design problems.