On a non-standard conjugation problem for elliptic equations

We investigate the regular solvability of the conjugation problem for elliptic equations with non-standard boundary conditions and sewing conditions on the plane $x = 0$. Let $Q$ be a parallelepiped. On the bottom of $Q$ we give a boundary condition for $u(x, t, a)$ in the part where $x>0$ and for $u_t(x, t, a)$ in the part where $x<0$. On the plane $x=0$ these conditions “intertwist”, so on the top of $Q$ we give a boundary condition for $u(x, t, a)$ in the part where $x<0$ and for $u_t(x, t, a)$ in the part where $x > 0$. Combining the regularization method and natural parameter continuation, we prove the uniqueness and existence theorems for regular solutions of this non-standard conjugation problem.