Abstract
In this paper, we establish the discreteness of transmission eigenvalues for Maxwell's equations. More precisely, we show that the spectrum of the transmission eigenvalue problem is discrete if the electromagnetic parameters epsilon, mu, (epsilon) over cap, (mu) over cap in the equations characterizing the inhomogeneity and background are smooth in some neighborhood of the boundary and isotropic on the boundary, and satisfy the conditions epsilon not equal (epsilon) over cap, mu not equal (mu) over cap, and epsilon/mu not equal (epsilon) over cap/(mu) over cap on the boundary. These are quite general assumptions on the coefficients, which are easy to check. To our knowledge, our paper is the first to establish discreteness of transmission eigenvalues for Maxwell's equations without assuming any restrictions on the sign combination of the contrasts epsilon-(epsilon) over cap and mu-(mu) over cap near the boundary and allowing for all the electromagnetic parameters to be inhomogeneous and anisotropic, except for on the boundary where they are isotropic but not necessarily constant as is often assumed in the literature.