Existence of weak solutions to refraction problems in Negative Refractive Index Materials

We prove existence of weak solutions to certain refraction problems in the setting of materials with negative refractive index, both in the near and far field. Solutions are obtained when the target measure is discrete, as well as when it is given by a general Radon measure. The far field problem is treated using techniques from optimal mass transport. As such a fully nonlinear PDE of Monge-Ampère type arises in this setting, and we calculate the coefficients explicitly. Finally, we produce an example in which the A3w condition holds but the A3s condition does not.