BOUNDARY VALUE PROBLEMS FOR THE 2ND-ORDER SEIBERG-WITTEN EQUATIONS CELSO MELCHIADES DORIA Received 8 June 2004 It is shown that the nonhomogeneous Dirichlet and Neuman problems for the 2nd-order Seiberg-Witten equation on a compact 4-manifold X 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 C of configuration space. The coercivity of the α -functional, when restricted into the α Coulomb subspace, imply the existence of a weak solution. The. | BOUNDARY VALUE PROBLEMS FOR THE 2ND-ORDER SEIBERG-WITTEN EQUATIONS CELSO MELCHIADES DORIA Received 8 June 2004 It is shown that the nonhomogeneous Dirichlet and Neuman problems for the 2nd-order Seiberg-Witten equation on a compact 4-manifold X admit a regular solution once the nonhomogeneous Palais-Smale condition GX 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 C of configuration space. The coercivity of the yWa-functional when restricted into the Coulomb subspace imply the existence of a weak solution. The regularity then follows from the boundedness of L -norms of spinor solutions and the gauge fixing lemma. 1. Introduction Let X be a compact smooth 4-manifold with nonempty boundary. In our context the Seiberg-Witten equations are the 2nd-order Euler-Lagrange equation of the functional defined in Definition . When the boundary is empty their variational aspects were first studied in 3 and the topological ones in 1 . Thus the main aim here is to obtain the existence of a solution to the nonhomogeneous equations whenever dX 0. The nonemptiness of the boundary inflicts boundary conditions on the problem. Classically this sort of problem is classified according to its boundary conditions in Dirichlet problem S or Neumann problem X . Originally the Seiberg-Witten equations were described in 8 as a pair of 1st-order PDE. The solutions of these equations were known as ff W a-monopoles and their main achievement were to shed light on the understanding of the 4-dimensional differential topology since new smooth invariants were defined by the topology of their moduli space of solutions moduli gauge group . In the same article Witten introduced a variational formulation for the equations and showed that its stable critical points turn out to be exactly the ff Wa-nionopoles. The variational .
