We consider an optimization problem related to elliptic PDEs of the form -\text{div}(a(x)\nabla u)=f − div ( a ( x ) ∇ u ) = f with Dirichlet boundary condition on a given domain \Omega Ω . The coefficient a(x) a ( x ) has to be determined, in a suitable given class of admissible choices, in order to optimize a given criterion. We first deal with the case when the cost is the so-called elastic compliance, and then we discuss the more general case when the problem is written as an optimal control problem.