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The transmission problem is a system of two second-order elliptic equationsof two unknowns equipped with the Cauchy data on the boundary. After fourdecades of research motivated by scattering theory, the spectral properties ofthis problem are now known to depend on a type of contrast between coefficientsnear the boundary. Previously, we established the discreteness of eigenvaluesfor a large class of anisotropic coefficients which is related to thecelebrated complementing conditions due to Agmon, Douglis, and Nirenberg. Inthis work, we establish the Weyl law for the eigenvalues and the completenessof the generalized eigenfunctions for this class of coefficients under anadditional mild assumption on the continuity of the coefficients. The analysisis new and based on the $L^p$ regularity theory for the transmission problemestablished here. It also involves a subtle application of the spectral theoryfor the Hilbert Schmidt operators. Our work extends largely known results inthe literature which are mainly devoted to the isotropic case with$C^\infty$-coefficients.

The Weyl law of transmission eigenvalues and the completeness of generalized transmission eigenfunctions