The enclosure method for a generalized anisotropic complex conductivity equation

We study how to apply the enclosure method to reconstruct an unknown inclusion within a medium in a domain in $R^n$ which satisfies the conductivity equation ∇ ⋅ ((σ ₀ + iɛ ₀)∇u) = 0 with σ ₀ and ɛ ₀ being real matrix functions. Motivated by some real world applications, we assume the unknown inclusion satisfies an equation of the more general form $\nabla \cdot ((\sigma +\text{i}\epsilon )\nabla u+\zeta \overline{\nabla u})=0$, where σ, ɛ, ζ are also real matrix functions. Due to the anisotropy, it is in general difficult to find complex geometric optics solutions. Therefore, we construct the oscillating decaying solutions, which is used to test whether a given half-space intersects the unknown inclusion or not. (paper)