Boundary value problems for the nd-order Seiberg-Witten equations
- PDF / 656,329 Bytes
- 19 Pages / 468 x 680 pts Page_size
- 14 Downloads / 230 Views
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 Ᏼ 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 ᏯCα of configuration space. The coercivity of the ᐃα -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 2.3. 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 ∂X = ∅. 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 (Ᏸ) or Neumann problem (ᏺ). Originally, the Seiberg-Witten equations were described in [8] as a pair of 1st-order PDE. The solutions of these equations were known as ᐃα -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 ᐃα -monopoles. The variational aspects of the ᐃα -equations were first explored in [3], where they proved that the functional satisfies the Palais-Smale condition and the solutions of the Euler-Lagrange (2nd-order) equations share the same important analytical properties as the ᐃα -monopoles. Therefore, it is natural to ask if the equations fit into a Morse-Bott-Smale theory, where the lower number of critical points Copyright © 2005 Hindawi Publishing Corporation Boundary Value Problems 2005:1 (2005) 73–91 DOI: 10.1155/BVP.2005.73
74
Boundary value problems for the ᐃα -equations
is the Betti number of the configuration space. The topology of the configuration space was described in [1]. Besides, if the SW-theory is a Morse theory, another natural question is to argue about the existence of a Morse-Smale-Witten complex, as in [6]. In the last question, the ᐃα -equations on manifolds endowed with tubular ends or boundary also demand attention. The analogy of the ᐃα -equation’s variational formulation, with the variational principle of the Ginzburg-Landau equation in superconductivity, further motivates the present study. 1.1. Spinc structure. The space of Spinc structur
Data Loading...
