An LP estimate for Maxwell's equations with source term

Abstract

In this Note we establish an interior L p -type estimate for the solutions of Maxwell's equations with source term in a domain filled with two different materials separated by a C 2 interface. Due to the singularity of the dielectric permittivity, the usual elliptic estimates cannot be applied directly. A special curl–div decomposition is introduced for the electric field to reduce the problem to an elliptic equation in divergence form with discontinuous coefficients. The potential theory analysis and the jump condition lead to the L p estimates which are superior to the straightforward Nash–Moser estimates. The reduction procedure is expected to be useful for numerical simulation. Such an estimate is crucial for solving nonlinear Maxwell's equations that arise for example in the modeling of nonlinear optics. To cite this article: G. Bao et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).