SIAM Journal on Mathematical Analysis, Volume 53, Issue 1, Page 888-913, January 2021.
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 $\varepsilon, \, \mu, \, \hat \varepsilon, \, \hat \mu$ 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 $\varepsilon \neq \hat \varepsilon$, $\mu \neq \hat \mu$, and $\varepsilon/ \mu \neq \hat \varepsilon/ \hat \mu$ 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 $\varepsilon-\hat \varepsilon$ and $\mu - \hat \mu$ 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.

中文翻译：

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麦克斯韦方程组传递本征值的离散性

SIAM数学分析杂志，第53卷，第1期，第888-913页，2021年1月。
在本文中，我们建立了麦克斯韦方程组透射本征值的离散性。更准确地说，我们表明，如果在表征不均匀性和背景的方程式中，电磁参数$ \ varepsilon，\，\ mu，\，\ hat \ varepsilon，\，\ hat \ mu $，则传输特征值问题的频谱是离散的在边界的某个邻域内是光滑的，边界上是各向同性的，并且满足条件$ \ varepsilon \ neq \ hat \ varepsilon $，$ \ mu \ neq \ hat \ mu $和$ \ varepsilon / \ mu \ neq \边界上的hat \ varepsilon / \ hat \ mu $。这些是关于系数的相当笼统的假设，很容易检查。据我们所知，我们的论文是第一个为麦克斯韦建立透射本征值离散性的论文。