Variational Aspects of the Seiberg-Witten Functional

The Seiberg-Witten equations that have recently found important applications for four-dimensional geometry are the Euler-Lagrange equations for a functional involving a connection $A$ on a line bundle $L$ and a section $\phi$ of another bundle $W^+$ constructed from $L$ and a spinor bundle on a given four-dimensional Riemannian manifold. We show the regularity of weak solutions and the Palais-Smale condition for this functional.