  This work deals with the problem of determining a non-homogeneous heat
conductivity profile in a steady-state heat conduction boundary-value problem
with mixed Dirichlet-Neumann boundary conditions over a bounded domain in
$\mathbb{R}^n$, from the knowledge of the state over the whole domain. We
develop a method based on a variational approach leading to an optimality
equation which is then projected into a finite dimensional space.
Discretization yields a linear although severely ill-posed equation which is
then regularized via appropriate ad-hoc penalizers resulting a in a generalized
Tikhonov-Phillips functional. No smoothness assumptions are imposed on the
conductivity. Numerical examples for the case in which the conductivity can
take only two prescribed values (a two-materials case) show that the approach
is able to produce very good reconstructions of the exact solution.
