Optimal Coefficients for Elliptic PDEs

We consider an optimization problem related to elliptic PDEs of the form

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.