On the Discreteness of Transmission Eigenvalues for the Maxwell Equations

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 $\eps, \, \mu, \, \heps, \, \hmu$ in the equations characterizing the inhomogeneity and background, are smooth in some neighborhood of the boundary, isotropic on the boundary, and satisfy the conditions $\eps \neq \heps$, $\mu \neq \hmu$, and $\eps/ \mu \neq \heps/ \hmu$ 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 $\eps-\heps$ and $\mu - \hmu$ 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 it is often assumed in the literature.