  We consider well-posedness of the boundary value problem in presence of an
inclusion with complex conductivity $k$. We first consider the transmission
problem in $\mathbb{R}^d$ and characterize solvability of the problem in terms
of the spectrum of the Neumann-Poincar\'e operator. We then deal with the
boundary value problem and show that the solution is bounded in its $H^1$-norm
uniformly in $k$ as long as $k$ is at some distance from a closed interval in
the negative real axis. We then show with an estimate that the solution depends
on $k$ in its $H^1$-norm Lipschitz continuously. We finally show that the
boundary perturbation formula in presence of a diametrically small inclusion is
valid uniformly in $k$ away from the closed interval mentioned before. The
results for the single inclusion case are extended to the case when there are
multiple inclusions with different complex conductivities: We first obtain a
complete characterization of solvability when inclusions consist of two
disjoint disks and then prove solvability and uniform estimates when imaginary
parts of conductivities have the same signs. The results are obtained using the
spectral property of the associated Neumann-Poincar\'e operator and the
spectral resolution.
