Boundary Value Problems
Volume 2005 (2005), Issue 1, Pages 73-91
doi:10.1155/BVP.2005.73
Abstract
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
ℋ 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 𝒞αℭ of configuration space. The coercivity of the 𝒮𝒲α-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.