  We investigate the continuity of boundary operators, such as the
Neumann-to-Dirichlet map, with respect to the coefficient matrices of the
underlying elliptic equations. We show that for nonsmooth coefficients the
correct notion of convergence is the one provided by $H$-convergence (or
$G$-convergence for symmetric matrices). We prove existence results for minimum
problems associated to variational methods used to solve the so-called inverse
conductivity problem, at least if we allow the conductivities to be
anisotropic. In the case of isotropic conductivities we show that on certain
occasions existence of a minimizer may fail.
