  In this paper, we study the Seiberg-Witten equations on the product R x Y,
where Y is a compact 3-manifold with boundary. Following the approach of
Salamon and Wehrheim in the instanton case, we impose Lagrangian boundary
conditions for the Seiberg- Witten equations. The resulting equations we obtain
constitute a nonlinear, nonlocal boundary value problem. We establish
regularity, compactness, and Fredholm properties for the Seiberg- Witten
equations supplied with Lagrangian boundary conditions arising from the
monopole spaces studied in [20]. This work therefore serves as an analytic
foundation for the construction of a monopole Floer theory for 3-manifolds with
boundary.
