The nonlocal boundary value problem with constant coefficients for the multidimensional mixed type equation of the first kind

In this paper the unique solvability and smoothness of generalized solution of a nonlocal boundary value problem with constant coefficients for the multidimensional mixed type equation of the first kind in Sobolev spaces $W_{2}^{l }(Q)$, ($2\le l$ is integer number), have been proved. First, the unique solvability of the generalized solution from space $W_{2}^{2 }(Q)$ has been studied. Further, the uniqueness of the generalized solution of nonlocal boundary value problem with constant coefficients for the multidimensional mixed type equation was proved by a priory estimates. For the proof of the existence of the generalized solution, we used method of "$\varepsilon$-regularization" together with Galerkin method. Precisely, first, we study regular solvability of the nonlocal boundary value problem for the multidimensional mixed type equation by functional analysis methods, i.e. we obtained necessary a priory estimates for the considered problems. Using these estimates we solve composite type equation, then by the theorem on weak compactness, we pass to the limit and deduce to the multidimensional mixed type equation of the first kind. At the end, smoothness of the generalized solution of the considered problems has been discussed.