Near Field Refraction Problem With Loss of Energy in Negative Refractive Index Material

This paper studies the near field refraction problem with loss of energy in negative refractive index material. Based on the relative refractive index $\kappa$, the analysis is categorized into two cases, namely $\kappa <-1$ and $-1<\kappa <0$. For each case, we give the definition of the refractor and discuss some crucial properties of it. The properties of Fresnel coefficients are also discussed. Based on these properties, the existence of the weak solution when the target measure is either discrete or a finite Radon measure are proved. Besides, the critical case $\kappa =-1$ is also discussed briefly at the end of this paper.