Abstract
It is shown that the nonhomogeneous Dirichlet and Neuman problems for thend-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.
Highlights
Let X be a compact smooth 4-manifold with nonempty boundary
The Seiberg-Witten equations are the 2nd-order Euler-Lagrange equation of the functional defined in Definition 2.3
The nonemptiness of the boundary inflicts boundary conditions on the problem. This sort of problem is classified according to its boundary conditions in Dirichlet problem (Ᏸ) or Neumann problem (ᏺ)
Summary
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 follows from the boundedness of L∞-norms of spinor solutions and the gauge fixing lemma
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