| The goal of this dissertation is to establish interior L p-type estimates for the solutions of Maxwell's equations with source term in a domain with two different materials separated by a interface. Such an estimate is employed to solve a nonlinear optics problem. The usual elliptic estimates cannot be applied directly, due to the singularity of the dielectric permittivity. A special curl-div decomposition is introduced for the electric field to reduce the problem to an elliptic equation in divergence form with jump coefficients. The potential analysis and the jump condition lead to the Lp estimates which are superior to the straightforward Nash-Moser estimates. The reduction procedure is expected to be useful for future numerical simulation. The problem arises in the modeling of surface enhanced second-harmonic generation of nonlinear diffractive optics in periodic structures (gratings). |
