Limiting absorption principle of Helmholtz equation with sign changing coefficients under periodic structure

Negative refractive index materials have attracted significant research attention due to their unique electromagnetic response characteristics. In this paper, we employ the complementing boundary condition to establish rigorous a priori estimates for the Helmholtz equation, from which the limiting absorption principle is analytically derived. Within this mathematical framework, we conclusively establish the well-posedness of the electromagnetic transmission problem at the interface between conventional materials and negative refractive index materials in two-dimensional periodic structures.