Boundary value problems for the nd-order Seiberg-Witten equations

It is shown that the nonhomogeneous Dirichlet and Neuman problems for the nd-order Seiberg-Witten equation on a compact -manifold admit a regular solution once the nonhomogeneous Palais-Smale condition is satisfied. The approach consists in applying the elliptic techniques to the variational setting of the Seiberg-Witten equation. The gauge invariance of the functional allows to restrict the problem to the Coulomb subspace of configuration space. The coercivity of the -functional, when restricted into the Coulomb subspace, imply the existence of a weak solution. The regularity then follows from the boundedness of -norms of spinor solutions and the gauge fixing lemma.