Reciprocity gap method for inclusions with variable conductivity

We extend the reciprocity gap approximation theorem proved by Di Cristo and Milan (2020) to the case of spatially varying conductivity coefficients. We consider conductivities of the form γ ( x ) = a ( x ) + b ( x ) χ D ( x ) where a , b ∈ L ∞ ( Ω ) satisfy uniform ellipticity conditions. We show that the fundamental dichotomy underlying the reciprocity gap method, based on bounded approximating sequences for sampling points inside the inclusion and blow-up outside, remains valid without assuming piecewise constant conductivity. The proof relies on variational arguments and weighted transmission conditions and does not require additional regularity of the inclusion.