  We investigate some scalar transmission problems between a classical positive
material and a negative one, whose physical coefficients are negative. First,
we consider cases where the negative inclusion is a disk in 2d and a ball in
3d. Thanks to asymptotics of Bessel functions (validated numerically), we show
well-posedness but with some possible loses of regularity of the solution
compared to the classical case of transmission problems between two positive
materials. Noticing that the curvature plays a central role, we then explore
the case of flat interfaces in the context of waveguides. In this case, the
transmission problem can also have some loses of regularity, or even be
ill-posed (kernel of infinite dimension).
