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We investigate the continuity of boundary operators, such as theNeumann-to-Dirichlet map, with respect to the coefficient matrices of theunderlying elliptic equations. We show that for nonsmooth coefficients thecorrect notion of convergence is the one provided by $H$-convergence (or$G$-convergence for symmetric matrices). We prove existence results for minimumproblems associated to variational methods used to solve the so-called inverseconductivity problem, at least if we allow the conductivities to beanisotropic. In the case of isotropic conductivities we show that on certainoccasions existence of a minimizer may fail.

Continuity properties of Neumann-to-Dirichlet maps with respect to the $H$-convergence of the coefficient matrices