  We consider first-order elliptic differential operators acting on vector
bundles over smooth manifolds with smooth boundary, which is permitted to be
noncompact. Under very mild assumptions, we obtain a regularity theory for
sections in the maximal domain. Under additional geometric assumptions, and
assumptions on an adapted boundary operator, we obtain a trace theorem on the
maximal domain. This allows us to systematically study both local and nonlocal
boundary conditions. In particular, the Atiyah-Patodi-Singer boundary condition
occurs as a special case. Furthermore, we study contexts which induce
semi-Fredholm and Fredholm extensions.
