We derive exact reconstruction methods for cracks consisting of unions of Lipschitz hypersurfaces in the context of Calder\'on's inverse conductivity problem. Our first method obtains upper bounds for the unknown cracks, bounds that can be shrunk to obtain the exact crack locations upon verifying certain operator inequalities for differences of the local Neumann-to-Dirichlet maps. This method can simultaneously handle perfectly insulating and perfectly conducting cracks, and it appears to be the first rigorous reconstruction method capable of this. Our second method assumes that only perfectly insulating cracks or only perfectly conducting cracks are present. Once more using operator inequalities, this method generates approximate cracks that are guaranteed to be subsets of the unknown cracks that are being reconstructed.

翻译：我们推导了在Calderón逆电导率问题中由Lipschitz超曲面并集构成的裂纹的精确重建方法。第一种方法通过验证局部Neumann-to-Dirichlet映射差分的特定算子不等式，获得未知裂纹的上界，该上界可收缩至精确裂纹位置。该方法能同时处理完美绝缘裂纹与完美导电裂纹，是首个具备此能力的严格重建方法。第二种方法假设仅存在完美绝缘裂纹或仅存在完美导电裂纹。再次利用算子不等式，该方法生成的近似裂纹保证是待重建未知裂纹的子集。