On Fredholm solvability of first boundary value problem for mixed-type second-order equation with spectral parameter

We study the first boundary problem for the mixed-type second-order equation with a spectral parameter in a cylindrical domain in $R^{n+1}$. Previously, for the mixed-type second-order equations some results were received only in two-dimensional domains. V. N. Vragov was the first to propose a well-posed statement of a boundary problem for the mixed-type equations. One of the well-posedness conditions is non-negativity of the spectral parameter. Here we analyze the case of complex spectral parameter and receive a priori estimates under certain conditions, using which an existence and uniqueness theorem is proved for the first boundary problem in the energy class. Also, we obtain sufficient conditions for the Fredholm solvability in the energy class.