Abstract: A fundamental problem in finding the best shape of a domain, in the sense that it minimizes a certain objective function is the calculus of the shape derivative of the objective function.

With this goal in mind, we present an elliptic shape optimization we start by introducing the model problem in conductivity and linearized elasticity, the conditions that assure the existence and uniqueness of solution for it's variational problem. We then consider a set of admissible domains which leads to the definition of shape derivative and we present some examples of existence of the optimal domain.

In this work we present the essential ideas to deduce the shape derivative of a class of objective functions following the method of F. Murat and J. Simon. Then we show a numerical example in shape optimization of a two dimensional cantilever in linear elasticity using finite elements.