The reciprocity gap method with Neumann boundary measurements

We consider the reciprocity gap method for the inverse inclusion problem in electrical conductivity when Neumann boundary measurements are prescribed. The inclusion is embedded in a known, possibly heterogeneous, background conductivity a(x) and is modeled by a constant relative conductivity contrast. We show that the fundamental structural property underlying the method, namely the bounded/blow-up dichotomy of approximating sequences, persists in this setting. We derive a weighted reciprocity gap identity and establish a representation formula for singular solutions in terms of an interior Neumann-to-Dirichlet map associated with the background conductivity. The harmonic test functions of the constant-background formulation are replaced by solutions of the corresponding background equation, and a density property for their boundary traces provides the approximation mechanism. We prove that sampling points inside the inclusion admit bounded approximating sequences of background solutions, whereas exterior points necessarily generate blow-up behavior. A numerical illustration for a nonconstant background conductivity confirms the qualitative interior/exterior separation predicted by the theoretical result.