  We study a transmission problem for the time harmonic Maxwell's equations
between a classical positive material and a so-called negative index material
in which both the permittivity $\varepsilon$ and the permeability $\mu$ take
negative values. Additionally, we assume that the interface between the two
domains is smooth everywhere except at a point where it coincides locally with
a conical tip. In this context, it is known that for certain critical values of
the contrasts in $\varepsilon$ and in $\mu$, the corresponding scalar operators
are not of Fredholm type in the usual $H^1$ spaces. In this work, we show that
in these situations, the Maxwell's equations are not well-posed in the
classical $L^2$ framework due to existence of hypersingular fields which are of
infinite energy at the tip. By combining the $\mathrm{T}$-coercivity approach
and the Kondratiev theory, we explain how to construct new functional
frameworks to recover well-posedness of the Maxwell's problem. We also explain
how to select the setting which is consistent with the limiting absorption
principle. From a technical point of view, the fields as well as their curls
decompose as the sum of an explicit singular part, related to the black hole
singularities of the scalar operators, and a smooth part belonging to some
weighted spaces. The analysis we propose rely in particular on the proof of new
key results of scalar and vector potential representations of singular fields.
