The Weyl law of transmission eigenvalues and the completeness of generalized transmission eigenfunctions without complementing conditions

The transmission eigenvalue problem is a system of two second-order elliptic equations of two unknowns equipped with the Cauchy data on the boundary. In this work, we establish the Weyl law for the eigenvalues and the completeness of the generalized eigenfunctions for a system without complementing conditions, i.e., the two equations of the system have the same coefficients for the second order terms, and thus being degenerate. These coefficients are allowed to be anisotropic and are assumed to be of class $C^2$. One of the keys of the analysis is to establish the well-posedness and the regularity in $L^p$-scale for such a system. As a result, we largely extend and rediscover known results for which the coefficients for the second order terms are required to be isotropic and of class $C^\infty$ using a new approach.